Understanding Exponents and Radicals In mathematics, exponents and radicals are two fundamental concepts that are closely related. Exponents represent repeated multiplication, while radicals represent the inverse operation of raising a number to a power. In this article, we will explore how to rewrite exponents in radical form, focusing on the given problems: $(7 m)^{\frac{3}{4}}$ and $(3 x)^{\frac{-5}{7}}$.
Rewriting Exponents in Radical Form: A Step-by-Step Guide To rewrite an exponent in radical form, we need to follow a simple step-by-step process. Here's how to do it:
Determine the power : Identify the exponent and determine its power. In the given problems, the exponents are 3 4 \frac{3}{4} 4 3 − 5 7 \frac{-5}{7} 7 − 5 Rewrite the exponent as a power : Rewrite the exponent as a power of the base. For example, if the exponent is 3 4 \frac{3}{4} 4 3 4 3 4 4^{\frac{3}{4}} 4 4 3 Take the root : Take the root of the power. In this case, we need to take the fourth root of the power.
Rewriting $(7 m)^{\frac{3}{4}}$ in Radical Form Let's apply the step-by-step process to rewrite $(7 m)^{\frac{3}{4}}$ in radical form.
Determine the power : The exponent is 3 4 \frac{3}{4} 4 3 Rewrite the exponent as a power : We can rewrite the exponent as 4 3 4 4^{\frac{3}{4}} 4 4 3 Take the root : We need to take the fourth root of the power.
Using the above steps, we can rewrite $(7 m)^{\frac{3}{4}}$ in radical form as:
( 7 m ) 3 4 \sqrt[4]{(7 m)^3}
4 ( 7 m ) 3
This is option B .
Rewriting $(3 x)^{\frac{-5}{7}}$ in Radical Form Now, let's apply the step-by-step process to rewrite $(3 x)^{\frac{-5}{7}}$ in radical form.
Determine the power : The exponent is − 5 7 \frac{-5}{7} 7 − 5 Rewrite the exponent as a power : We can rewrite the exponent as 7 − 5 7 7^{\frac{-5}{7}} 7 7 − 5 Take the root : We need to take the seventh root of the power.
Using the above steps, we can rewrite $(3 x)^{\frac{-5}{7}}$ in radical form as:
( 3 x ) − 5 7 \sqrt[7]{(3 x)^{-5}}
7 ( 3 x ) − 5
However, we can simplify this expression further by applying the rule of negative exponents:
( 3 x ) − 5 7 = 1 ( 3 x ) 5 7 \sqrt[7]{(3 x)^{-5}} = \frac{1}{\sqrt[7]{(3 x)^5}}
7 ( 3 x ) − 5 = 7 ( 3 x ) 5 1
This is not an option, but we can rewrite it as:
1 ( 3 x ) 5 7 = 1 3 5 x 5 7 = 1 3 5 7 x 5 7 \frac{1}{\sqrt[7]{(3 x)^5}} = \frac{1}{\sqrt[7]{3^5 x^5}} = \frac{1}{3^{\frac{5}{7}} x^{\frac{5}{7}}}
7 ( 3 x ) 5 1 = 7 3 5 x 5 1 = 3 7 5 x 7 5 1
However, we can simplify this expression further by applying the rule of negative exponents:
1 3 5 7 x 5 7 = 1 3 5 7 x 5 7 ⋅ 3 2 7 x 2 7 3 2 7 x 2 7 = 3 2 7 x 2 7 3 7 7 x 7 7 = 3 2 7 x 2 7 3 x \frac{1}{3^{\frac{5}{7}} x^{\frac{5}{7}}} = \frac{1}{3^{\frac{5}{7}} x^{\frac{5}{7}}} \cdot \frac{3^{\frac{2}{7}} x^{\frac{2}{7}}}{3^{\frac{2}{7}} x^{\frac{2}{7}}} = \frac{3^{\frac{2}{7}} x^{\frac{2}{7}}}{3^{\frac{7}{7}} x^{\frac{7}{7}}} = \frac{3^{\frac{2}{7}} x^{\frac{2}{7}}}{3 x}
3 7 5 x 7 5 1 = 3 7 5 x 7 5 1 ⋅ 3 7 2 x 7 2 3 7 2 x 7 2 = 3 7 7 x 7 7 3 7 2 x 7 2 = 3 x 3 7 2 x 7 2
However, we can simplify this expression further by applying the rule of negative exponents:
3 2 7 x 2 7 3 x = 3 2 7 x 2 7 3 x ⋅ 3 − 5 7 x − 5 7 3 − 5 7 x − 5 7 = 3 2 7 − 5 7 x 2 7 − 5 7 3 − 5 7 x − 5 7 = 3 − 3 7 x − 3 7 3 − 5 7 x − 5 7 \frac{3^{\frac{2}{7}} x^{\frac{2}{7}}}{3 x} = \frac{3^{\frac{2}{7}} x^{\frac{2}{7}}}{3 x} \cdot \frac{3^{-\frac{5}{7}} x^{-\frac{5}{7}}}{3^{-\frac{5}{7}} x^{-\frac{5}{7}}} = \frac{3^{\frac{2}{7} - \frac{5}{7}} x^{\frac{2}{7} - \frac{5}{7}}}{3^{-\frac{5}{7}} x^{-\frac{5}{7}}} = \frac{3^{-\frac{3}{7}} x^{-\frac{3}{7}}}{3^{-\frac{5}{7}} x^{-\frac{5}{7}}}
3 x 3 7 2 x 7 2 = 3 x 3 7 2 x 7 2 ⋅ 3 − 7 5 x − 7 5 3 − 7 5 x − 7 5 = 3 − 7 5 x − 7 5 3 7 2 − 7 5 x 7 2 − 7 5 = 3 − 7 5 x − 7 5 3 − 7 3 x − 7 3
However, we can simplify this expression further by applying the rule of negative exponents:
3 − 3 7 x − 3 7 3 − 5 7 x − 5 7 = 3 − 3 7 x − 3 7 3 − 5 7 x − 5 7 ⋅ 3 2 7 x 2 7 3 2 7 x 2 7 = 3 − 3 7 + 2 7 x − 3 7 + 2 7 3 2 7 x 2 7 = 3 − 1 7 x − 1 7 3 2 7 x 2 7 \frac{3^{-\frac{3}{7}} x^{-\frac{3}{7}}}{3^{-\frac{5}{7}} x^{-\frac{5}{7}}} = \frac{3^{-\frac{3}{7}} x^{-\frac{3}{7}}}{3^{-\frac{5}{7}} x^{-\frac{5}{7}}} \cdot \frac{3^{\frac{2}{7}} x^{\frac{2}{7}}}{3^{\frac{2}{7}} x^{\frac{2}{7}}} = \frac{3^{-\frac{3}{7} + \frac{2}{7}} x^{-\frac{3}{7} + \frac{2}{7}}}{3^{\frac{2}{7}} x^{\frac{2}{7}}} = \frac{3^{-\frac{1}{7}} x^{-\frac{1}{7}}}{3^{\frac{2}{7}} x^{\frac{2}{7}}}
3 − 7 5 x − 7 5 3 − 7 3 x − 7 3 = 3 − 7 5 x − 7 5 3 − 7 3 x − 7 3 ⋅ 3 7 2 x 7 2 3 7 2 x 7 2 = 3 7 2 x 7 2 3 − 7 3 + 7 2 x − 7 3 + 7 2 = 3 7 2 x 7 2 3 − 7 1 x − 7 1
However, we can simplify this expression further by applying the rule of negative exponents:
3 − 1 7 x − 1 7 3 2 7 x 2 7 = 3 − 1 7 x − 1 7 3 2 7 x 2 7 ⋅ 3 − 2 7 x − 2 7 3 − 2 7 x − 2 7 = 3 − 1 7 − 2 7 x − 1 7 − 2 7 3 − 2 7 x − 2 7 = 3 − 3 7 x − 3 7 3 − 2 7 x − 2 7 \frac{3^{-\frac{1}{7}} x^{-\frac{1}{7}}}{3^{\frac{2}{7}} x^{\frac{2}{7}}} = \frac{3^{-\frac{1}{7}} x^{-\frac{1}{7}}}{3^{\frac{2}{7}} x^{\frac{2}{7}}} \cdot \frac{3^{-\frac{2}{7}} x^{-\frac{2}{7}}}{3^{-\frac{2}{7}} x^{-\frac{2}{7}}} = \frac{3^{-\frac{1}{7} - \frac{2}{7}} x^{-\frac{1}{7} - \frac{2}{7}}}{3^{-\frac{2}{7}} x^{-\frac{2}{7}}} = \frac{3^{-\frac{3}{7}} x^{-\frac{3}{7}}}{3^{-\frac{2}{7}} x^{-\frac{2}{7}}}
3 7 2 x 7 2 3 − 7 1 x − 7 1 = 3 7 2 x 7 2 3 − 7 1 x − 7 1 ⋅ 3 − 7 2 x − 7 2 3 − 7 2 x − 7 2 = 3 − 7 2 x − 7 2 3 − 7 1 − 7 2 x − 7 1 − 7 2 = 3 − 7 2 x − 7 2 3 − 7 3 x − 7 3
However, we can simplify this expression further by applying the rule of negative exponents:
3 − 3 7 x − 3 7 3 − 2 7 x − 2 7 = 3 − 3 7 x − 3 7 3 − 2 7 x − 2 7 ⋅ 3 1 7 x 1 7 3 1 7 x 1 7 = 3 − 3 7 + 1 7 x − 3 7 + 1 7 3 1 7 x 1 7 = 3 − 2 7 x − 2 7 3 1 7 x 1 7 \frac{3^{-\frac{3}{7}} x^{-\frac{3}{7}}}{3^{-\frac{2}{7}} x^{-\frac{2}{7}}} = \frac{3^{-\frac{3}{7}} x^{-\frac{3}{7}}}{3^{-\frac{2}{7}} x^{-\frac{2}{7}}} \cdot \frac{3^{\frac{1}{7}} x^{\frac{1}{7}}}{3^{\frac{1}{7}} x^{\frac{1}{7}}} = \frac{3^{-\frac{3}{7} + \frac{1}{7}} x^{-\frac{3}{7} + \frac{1}{7}}}{3^{\frac{1}{7}} x^{\frac{1}{7}}} = \frac{3^{-\frac{2}{7}} x^{-\frac{2}{7}}}{3^{\frac{1}{7}} x^{\frac{1}{7}}}
3 − 7 2 x − 7 2 3 − 7 3 x − 7 3 = 3 − 7 2 x − 7 2 3 − 7 3 x − 7 3 ⋅ 3 7 1 x 7 1 3 7 1 x 7 1 = 3 7 1 x 7 1 3 − 7 3 + 7 1 x − 7 3 + 7 1 = 3 7 1 x 7 1 3 − 7 2 x − 7 2
However, we can simplify this expression further by applying the rule of negative exponents:
\frac{3^{-\frac{2}{7}} x^{-\frac{2}{7}}}{3^{\frac{1}{7}} x^{\frac{1}{7}}} = \frac{3^{-\frac{2}{7}} x^{-\frac{2}{7}}}{3^{\frac{1}{7}} x^{\frac{1}{7}}} \cdot \frac{3^{-\frac{1}{7}} x^{-\frac{1}{7}}}{3^{-\frac{1}{7}} x^{-\frac{1}{7}}} = \frac{3^{-\frac{2}{7} - \frac{1}{7}} x^{-\frac{2}{7} - \frac{1}{7}}}{3^{-\frac{1}{7}} x^{-\frac{1}{7}}} = \frac{3^{-\frac{3}{7}} x^{-\frac{3}{7}}}{3^{-\frac{<br/>
**Q&A: Rewriting Exponents in Radical Form**
=============================================
Q: What is the difference between exponents and radicals? A: Exponents and radicals are two fundamental concepts in mathematics that are closely related. Exponents represent repeated multiplication, while radicals represent the inverse operation of raising a number to a power.
Q: How do I rewrite an exponent in radical form? A: To rewrite an exponent in radical form, follow these steps:
Determine the power : Identify the exponent and determine its power.Rewrite the exponent as a power : Rewrite the exponent as a power of the base.Take the root : Take the root of the power.
Q: Can you give an example of rewriting an exponent in radical form? A: Let's say we want to rewrite the exponent ( 7 m ) 3 4 (7 m)^{\frac{3}{4}} ( 7 m ) 4 3
( 7 m ) 3 4 < / s p a n > < / p > < h 2 > < s t r o n g > Q : W h a t i s t h e r u l e f o r n e g a t i v e e x p o n e n t s ? < / s t r o n g > < / h 2 > < p > A : T h e r u l e f o r n e g a t i v e e x p o n e n t s s t a t e s t h a t < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u p > < m i > a < / m i > < m r o w > < m o > − < / m o > < m i > n < / m i > < / m r o w > < / m s u p > < m o > = < / m o > < m f r a c > < m n > 1 < / m n > < m s u p > < m i > a < / m i > < m i > n < / m i > < / m s u p > < / m f r a c > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > a − n = 1 a n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7713 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > a < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7713 e m ; " > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.1901 e m ; v e r t i c a l − a l i g n : − 0.345 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8451 e m ; " > < s p a n s t y l e = " t o p : − 2.655 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > a < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.5935 e m ; " > < s p a n s t y l e = " t o p : − 2.786 e m ; m a r g i n − r i g h t : 0.0714 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.5 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.394 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > . T h i s m e a n s t h a t a n e g a t i v e e x p o n e n t c a n b e r e w r i t t e n a s a f r a c t i o n w i t h t h e b a s e i n t h e d e n o m i n a t o r . < / p > < h 2 > < s t r o n g > Q : C a n y o u g i v e a n e x a m p l e o f a p p l y i n g t h e r u l e f o r n e g a t i v e e x p o n e n t s ? < / s t r o n g > < / h 2 > < p > A : L e t ′ s s a y w e w a n t t o r e w r i t e t h e e x p r e s s i o n < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m f r a c > < m n > 1 < / m n > < m r o o t > < m r o w > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 3 < / m n > < m i > x < / m i > < m s u p > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m n > 5 < / m n > < / m s u p > < / m r o w > < m n > 7 < / m n > < / m r o o t > < / m f r a c > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > 1 ( 3 x ) 5 7 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.6747 e m ; v e r t i c a l − a l i g n : − 0.8296 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8451 e m ; " > < s p a n s t y l e = " t o p : − 2.4642 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d s q r t m t i g h t " > < s p a n c l a s s = " r o o t " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8307 e m ; " > < s p a n s t y l e = " t o p : − 2.8704 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.5 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.0369 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3.4286 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3.4286 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " s t y l e = " p a d d i n g − l e f t : 1.19 e m ; " > < s p a n c l a s s = " m o p e n m t i g h t " > ( < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > x < / s p a n > < s p a n c l a s s = " m c l o s e m t i g h t " > < s p a n c l a s s = " m c l o s e m t i g h t " > ) < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7463 e m ; " > < s p a n s t y l e = " t o p : − 2.786 e m ; m a r g i n − r i g h t : 0.0714 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.5 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 5 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.0089 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3.4286 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l m t i g h t " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.5429 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.5429 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.4197 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.394 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8296 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > u s i n g t h e r u l e f o r n e g a t i v e e x p o n e n t s . W e c a n r e w r i t e i t a s : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m f r a c > < m n > 1 < / m n > < m r o o t > < m r o w > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 3 < / m n > < m i > x < / m i > < m s u p > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m n > 5 < / m n > < / m s u p > < / m r o w > < m n > 7 < / m n > < / m r o o t > < / m f r a c > < m o > = < / m o > < m f r a c > < m n > 1 < / m n > < m r o w > < m s u p > < m n > 3 < / m n > < m f r a c > < m n > 5 < / m n > < m n > 7 < / m n > < / m f r a c > < / m s u p > < m s u p > < m i > x < / m i > < m f r a c > < m n > 5 < / m n > < m n > 7 < / m n > < / m f r a c > < / m s u p > < / m r o w > < / m f r a c > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > 1 ( 3 x ) 5 7 = 1 3 5 7 x 5 7 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4514 e m ; v e r t i c a l − a l i g n : − 1.13 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.175 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " r o o t " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7002 e m ; " > < s p a n s t y l e = " t o p : − 2.878 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.5 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.935 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3.2 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3.2 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 1 e m ; " > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " > < s p a n s t y l e = " t o p : − 2.989 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 5 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.895 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3.2 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 1.02 e m ; h e i g h t : 1.28 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.28 e m " v i e w B o x = " 004000001296 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 263 , 681 c 0.7 , 0 , 18 , 39.7 , 52 , 119 c 34 , 79.3 , 68.167 , 158.7 , 102.5 , 238 c 34.3 , 79.3 , 51.8 , 119.3 , 52.5 , 120 c 340 , − 704.7 , 510.7 , − 1060.3 , 512 , − 1067 l 0 − 0 c 4.7 , − 7.3 , 11 , − 11 , 19 , − 11 H 40000 v 40 H 1012.3 s − 271.3 , 567 , − 271.3 , 567 c − 38.7 , 80.7 , − 84 , 175 , − 136 , 283 c − 52 , 108 , − 89.167 , 185.3 , − 111.5 , 232 c − 22.3 , 46.7 , − 33.8 , 70.3 , − 34.5 , 71 c − 4.7 , 4.7 , − 12.3 , 7 , − 23 , 7 s − 12 , − 1 , − 12 , − 1 s − 109 , − 253 , − 109 , − 253 c − 72.7 , − 168 , − 109.3 , − 252 , − 110 , − 252 c − 10.7 , 8 , − 22 , 16.7 , − 34 , 26 c − 22 , 17.3 , − 33.3 , 26 , − 34 , 26 s − 26 , − 26 , − 26 , − 26 s 76 , − 59 , 76 , − 59 s 76 , − 60 , 76 , − 60 z M 100180 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.305 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.13 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.151 e m ; v e r t i c a l − a l i g n : − 0.8296 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.1704 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " > < s p a n s t y l e = " t o p : − 3.3486 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 5 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " > < s p a n s t y l e = " t o p : − 3.3486 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 5 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8296 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 2 > < s t r o n g > Q : H o w d o I s i m p l i f y a n e x p r e s s i o n w i t h a n e g a t i v e e x p o n e n t ? < / s t r o n g > < / h 2 > < p > A : T o s i m p l i f y a n e x p r e s s i o n w i t h a n e g a t i v e e x p o n e n t , f o l l o w t h e s e s t e p s : < / p > < o l > < l i > < s t r o n g > A p p l y t h e r u l e f o r n e g a t i v e e x p o n e n t s < / s t r o n g > : R e w r i t e t h e n e g a t i v e e x p o n e n t a s a f r a c t i o n w i t h t h e b a s e i n t h e d e n o m i n a t o r . < / l i > < l i > < s t r o n g > S i m p l i f y t h e e x p r e s s i o n < / s t r o n g > : S i m p l i f y t h e r e s u l t i n g e x p r e s s i o n b y c o m b i n i n g l i k e t e r m s . < / l i > < / o l > < h 2 > < s t r o n g > Q : C a n y o u g i v e a n e x a m p l e o f s i m p l i f y i n g a n e x p r e s s i o n w i t h a n e g a t i v e e x p o n e n t ? < / s t r o n g > < / h 2 > < p > A : L e t ′ s s a y w e w a n t t o s i m p l i f y t h e e x p r e s s i o n < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m f r a c > < m r o w > < m s u p > < m n > 3 < / m n > < m r o w > < m o > − < / m o > < m f r a c > < m n > 3 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < m s u p > < m i > x < / m i > < m r o w > < m o > − < / m o > < m f r a c > < m n > 3 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < / m r o w > < m r o w > < m s u p > < m n > 3 < / m n > < m r o w > < m o > − < / m o > < m f r a c > < m n > 2 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < m s u p > < m i > x < / m i > < m r o w > < m o > − < / m o > < m f r a c > < m n > 2 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < / m r o w > < / m f r a c > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > 3 − 3 7 x − 3 7 3 − 2 7 x − 2 7 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.9183 e m ; v e r t i c a l − a l i g n : − 0.6671 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.2511 e m ; " > < s p a n s t y l e = " t o p : − 2.3329 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.2245 e m ; " > < s p a n s t y l e = " t o p : − 3.4878 e m ; m a r g i n − r i g h t : 0.0714 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 1 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.0314 e m ; " > < s p a n s t y l e = " t o p : − 2.468 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.387 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.532 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 1 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.2245 e m ; " > < s p a n s t y l e = " t o p : − 3.4878 e m ; m a r g i n − r i g h t : 0.0714 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 1 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.0314 e m ; " > < s p a n s t y l e = " t o p : − 2.468 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.387 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.532 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 1 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.394 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.2245 e m ; " > < s p a n s t y l e = " t o p : − 3.4878 e m ; m a r g i n − r i g h t : 0.0714 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 1 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.0314 e m ; " > < s p a n s t y l e = " t o p : − 2.468 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.387 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.532 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 1 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.2245 e m ; " > < s p a n s t y l e = " t o p : − 3.4878 e m ; m a r g i n − r i g h t : 0.0714 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 1 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.0314 e m ; " > < s p a n s t y l e = " t o p : − 2.468 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.387 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.532 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 1 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.6671 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > . F o l l o w i n g t h e s t e p s a b o v e , w e c a n s i m p l i f y i t a s : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m f r a c > < m r o w > < m s u p > < m n > 3 < / m n > < m r o w > < m o > − < / m o > < m f r a c > < m n > 3 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < m s u p > < m i > x < / m i > < m r o w > < m o > − < / m o > < m f r a c > < m n > 3 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < / m r o w > < m r o w > < m s u p > < m n > 3 < / m n > < m r o w > < m o > − < / m o > < m f r a c > < m n > 2 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < m s u p > < m i > x < / m i > < m r o w > < m o > − < / m o > < m f r a c > < m n > 2 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < / m r o w > < / m f r a c > < m o > = < / m o > < m f r a c > < m r o w > < m s u p > < m n > 3 < / m n > < m r o w > < m o > − < / m o > < m f r a c > < m n > 3 < / m n > < m n > 7 < / m n > < / m f r a c > < m o > + < / m o > < m f r a c > < m n > 2 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < m s u p > < m i > x < / m i > < m r o w > < m o > − < / m o > < m f r a c > < m n > 3 < / m n > < m n > 7 < / m n > < / m f r a c > < m o > + < / m o > < m f r a c > < m n > 2 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < / m r o w > < m r o w > < m s u p > < m n > 3 < / m n > < m f r a c > < m n > 2 < / m n > < m n > 7 < / m n > < / m f r a c > < / m s u p > < m s u p > < m i > x < / m i > < m f r a c > < m n > 2 < / m n > < m n > 7 < / m n > < / m f r a c > < / m s u p > < / m r o w > < / m f r a c > < m o > = < / m o > < m f r a c > < m r o w > < m s u p > < m n > 3 < / m n > < m r o w > < m o > − < / m o > < m f r a c > < m n > 1 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < m s u p > < m i > x < / m i > < m r o w > < m o > − < / m o > < m f r a c > < m n > 1 < / m n > < m n > 7 < / m n > < / m f r a c > < / m r o w > < / m s u p > < / m r o w > < m r o w > < m s u p > < m n > 3 < / m n > < m f r a c > < m n > 2 < / m n > < m n > 7 < / m n > < / m f r a c > < / m s u p > < m s u p > < m i > x < / m i > < m f r a c > < m n > 2 < / m n > < m n > 7 < / m n > < / m f r a c > < / m s u p > < / m r o w > < / m f r a c > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > 3 − 3 7 x − 3 7 3 − 2 7 x − 2 7 = 3 − 3 7 + 2 7 x − 3 7 + 2 7 3 2 7 x 2 7 = 3 − 1 7 x − 1 7 3 2 7 x 2 7 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4606 e m ; v e r t i c a l − a l i g n : − 0.8296 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.631 e m ; " > < s p a n s t y l e = " t o p : − 2.1704 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " > < s p a n s t y l e = " t o p : − 3.3486 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " > < s p a n s t y l e = " t o p : − 3.3486 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " > < s p a n s t y l e = " t o p : − 3.363 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " > < s p a n s t y l e = " t o p : − 3.363 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8296 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4606 e m ; v e r t i c a l − a l i g n : − 0.8296 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.631 e m ; " > < s p a n s t y l e = " t o p : − 2.1704 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " > < s p a n s t y l e = " t o p : − 3.3486 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " > < s p a n s t y l e = " t o p : − 3.3486 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " > < s p a n s t y l e = " t o p : − 3.363 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < s p a n c l a s s = " m b i n m t i g h t " > + < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " > < s p a n s t y l e = " t o p : − 3.363 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < s p a n c l a s s = " m b i n m t i g h t " > + < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8296 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4606 e m ; v e r t i c a l − a l i g n : − 0.8296 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.631 e m ; " > < s p a n s t y l e = " t o p : − 2.1704 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " > < s p a n s t y l e = " t o p : − 3.3486 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " > < s p a n s t y l e = " t o p : − 3.3486 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " > < s p a n s t y l e = " t o p : − 3.363 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " > < s p a n s t y l e = " t o p : − 3.363 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : − 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 7 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e m t i g h t " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t − s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8296 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 2 > < s t r o n g > Q : W h a t a r e s o m e c o m m o n m i s t a k e s t o a v o i d w h e n r e w r i t i n g e x p o n e n t s i n r a d i c a l f o r m ? < / s t r o n g > < / h 2 > < p > A : S o m e c o m m o n m i s t a k e s t o a v o i d w h e n r e w r i t i n g e x p o n e n t s i n r a d i c a l f o r m i n c l u d e : < / p > < u l > < l i > < s t r o n g > N o t f o l l o w i n g t h e o r d e r o f o p e r a t i o n s < / s t r o n g > : M a k e s u r e t o f o l l o w t h e o r d e r o f o p e r a t i o n s ( P E M D A S ) w h e n r e w r i t i n g e x p o n e n t s i n r a d i c a l f o r m . < / l i > < l i > < s t r o n g > N o t s i m p l i f y i n g t h e e x p r e s s i o n < / s t r o n g > : M a k e s u r e t o s i m p l i f y t h e r e s u l t i n g e x p r e s s i o n b y c o m b i n i n g l i k e t e r m s . < / l i > < l i > < s t r o n g > N o t a p p l y i n g t h e r u l e f o r n e g a t i v e e x p o n e n t s < / s t r o n g > : M a k e s u r e t o a p p l y t h e r u l e f o r n e g a t i v e e x p o n e n t s w h e n r e w r i t i n g e x p r e s s i o n s w i t h n e g a t i v e e x p o n e n t s . < / l i > < / u l > < h 2 > < s t r o n g > Q : H o w c a n I p r a c t i c e r e w r i t i n g e x p o n e n t s i n r a d i c a l f o r m ? < / s t r o n g > < / h 2 > < p > A : Y o u c a n p r a c t i c e r e w r i t i n g e x p o n e n t s i n r a d i c a l f o r m b y w o r k i n g t h r o u g h e x e r c i s e s a n d p r o b l e m s t h a t i n v o l v e r e w r i t i n g e x p o n e n t s i n r a d i c a l f o r m . Y o u c a n a l s o t r y u s i n g o n l i n e r e s o u r c e s o r m a t h s o f t w a r e t o h e l p y o u p r a c t i c e a n d l e a r n . < / p > \sqrt[4]{(7 m)^3}
</span></p>
<h2><strong>Q: What is the rule for negative exponents?</strong></h2>
<p>A: The rule for negative exponents states that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><msup><mi>a</mi><mi>n</mi></msup></mfrac></mrow><annotation encoding="application/x-tex">a^{-n} = \frac{1}{a^n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7713em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7713em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5935em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. This means that a negative exponent can be rewritten as a fraction with the base in the denominator.</p>
<h2><strong>Q: Can you give an example of applying the rule for negative exponents?</strong></h2>
<p>A: Let's say we want to rewrite the expression <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mroot><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><msup><mo stretchy="false">)</mo><mn>5</mn></msup></mrow><mn>7</mn></mroot></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{\sqrt[7]{(3 x)^5}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.6747em;vertical-align:-0.8296em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.4642em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8307em;"><span style="top:-2.8704em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0369em;"><span class="svg-align" style="top:-3.4286em;"><span class="pstrut" style="height:3.4286em;"></span><span class="mord mtight" style="padding-left:1.19em;"><span class="mopen mtight">(</span><span class="mord mtight">3</span><span class="mord mathnormal mtight">x</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span><span style="top:-3.0089em;"><span class="pstrut" style="height:3.4286em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.5429em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.5429em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4197em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8296em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> using the rule for negative exponents. We can rewrite it as:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mn>1</mn><mroot><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><msup><mo stretchy="false">)</mo><mn>5</mn></msup></mrow><mn>7</mn></mroot></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mn>3</mn><mfrac><mn>5</mn><mn>7</mn></mfrac></msup><msup><mi>x</mi><mfrac><mn>5</mn><mn>7</mn></mfrac></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{\sqrt[7]{(3 x)^5}} = \frac{1}{3^{\frac{5}{7}} x^{\frac{5}{7}}}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4514em;vertical-align:-1.13em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.175em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7002em;"><span style="top:-2.878em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mopen">(</span><span class="mord">3</span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120
c340,-704.7,510.7,-1060.3,512,-1067
l0 -0
c4.7,-7.3,11,-11,19,-11
H40000v40H1012.3
s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232
c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z
M1001 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.13em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.151em;vertical-align:-0.8296em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.1704em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9396em;"><span style="top:-3.3486em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9396em;"><span style="top:-3.3486em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8296em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<h2><strong>Q: How do I simplify an expression with a negative exponent?</strong></h2>
<p>A: To simplify an expression with a negative exponent, follow these steps:</p>
<ol>
<li><strong>Apply the rule for negative exponents</strong>: Rewrite the negative exponent as a fraction with the base in the denominator.</li>
<li><strong>Simplify the expression</strong>: Simplify the resulting expression by combining like terms.</li>
</ol>
<h2><strong>Q: Can you give an example of simplifying an expression with a negative exponent?</strong></h2>
<p>A: Let's say we want to simplify the expression <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msup><mn>3</mn><mrow><mo>−</mo><mfrac><mn>3</mn><mn>7</mn></mfrac></mrow></msup><msup><mi>x</mi><mrow><mo>−</mo><mfrac><mn>3</mn><mn>7</mn></mfrac></mrow></msup></mrow><mrow><msup><mn>3</mn><mrow><mo>−</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mrow></msup><msup><mi>x</mi><mrow><mo>−</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mrow></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{3^{-\frac{3}{7}} x^{-\frac{3}{7}}}{3^{-\frac{2}{7}} x^{-\frac{2}{7}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.9183em;vertical-align:-0.6671em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2511em;"><span style="top:-2.3329em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2245em;"><span style="top:-3.4878em;margin-right:0.0714em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size1 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0314em;"><span style="top:-2.468em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight"><span class="mord mtight">7</span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.387em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.532em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size1 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2245em;"><span style="top:-3.4878em;margin-right:0.0714em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size1 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0314em;"><span style="top:-2.468em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight"><span class="mord mtight">7</span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.387em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.532em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size1 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2245em;"><span style="top:-3.4878em;margin-right:0.0714em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size1 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0314em;"><span style="top:-2.468em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight"><span class="mord mtight">7</span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.387em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.532em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size1 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2245em;"><span style="top:-3.4878em;margin-right:0.0714em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size1 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0314em;"><span style="top:-2.468em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight"><span class="mord mtight">7</span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.387em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.532em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size1 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6671em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. Following the steps above, we can simplify it as:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><msup><mn>3</mn><mrow><mo>−</mo><mfrac><mn>3</mn><mn>7</mn></mfrac></mrow></msup><msup><mi>x</mi><mrow><mo>−</mo><mfrac><mn>3</mn><mn>7</mn></mfrac></mrow></msup></mrow><mrow><msup><mn>3</mn><mrow><mo>−</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mrow></msup><msup><mi>x</mi><mrow><mo>−</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mn>3</mn><mrow><mo>−</mo><mfrac><mn>3</mn><mn>7</mn></mfrac><mo>+</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mrow></msup><msup><mi>x</mi><mrow><mo>−</mo><mfrac><mn>3</mn><mn>7</mn></mfrac><mo>+</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mrow></msup></mrow><mrow><msup><mn>3</mn><mfrac><mn>2</mn><mn>7</mn></mfrac></msup><msup><mi>x</mi><mfrac><mn>2</mn><mn>7</mn></mfrac></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mn>3</mn><mrow><mo>−</mo><mfrac><mn>1</mn><mn>7</mn></mfrac></mrow></msup><msup><mi>x</mi><mrow><mo>−</mo><mfrac><mn>1</mn><mn>7</mn></mfrac></mrow></msup></mrow><mrow><msup><mn>3</mn><mfrac><mn>2</mn><mn>7</mn></mfrac></msup><msup><mi>x</mi><mfrac><mn>2</mn><mn>7</mn></mfrac></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{3^{-\frac{3}{7}} x^{-\frac{3}{7}}}{3^{-\frac{2}{7}} x^{-\frac{2}{7}}} = \frac{3^{-\frac{3}{7} + \frac{2}{7}} x^{-\frac{3}{7} + \frac{2}{7}}}{3^{\frac{2}{7}} x^{\frac{2}{7}}} = \frac{3^{-\frac{1}{7}} x^{-\frac{1}{7}}}{3^{\frac{2}{7}} x^{\frac{2}{7}}}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4606em;vertical-align:-0.8296em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.631em;"><span style="top:-2.1704em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9396em;"><span style="top:-3.3486em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9396em;"><span style="top:-3.3486em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8296em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4606em;vertical-align:-0.8296em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.631em;"><span style="top:-2.1704em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9396em;"><span style="top:-3.3486em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9396em;"><span style="top:-3.3486em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8296em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4606em;vertical-align:-0.8296em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.631em;"><span style="top:-2.1704em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9396em;"><span style="top:-3.3486em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9396em;"><span style="top:-3.3486em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8296em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<h2><strong>Q: What are some common mistakes to avoid when rewriting exponents in radical form?</strong></h2>
<p>A: Some common mistakes to avoid when rewriting exponents in radical form include:</p>
<ul>
<li><strong>Not following the order of operations</strong>: Make sure to follow the order of operations (PEMDAS) when rewriting exponents in radical form.</li>
<li><strong>Not simplifying the expression</strong>: Make sure to simplify the resulting expression by combining like terms.</li>
<li><strong>Not applying the rule for negative exponents</strong>: Make sure to apply the rule for negative exponents when rewriting expressions with negative exponents.</li>
</ul>
<h2><strong>Q: How can I practice rewriting exponents in radical form?</strong></h2>
<p>A: You can practice rewriting exponents in radical form by working through exercises and problems that involve rewriting exponents in radical form. You can also try using online resources or math software to help you practice and learn.</p>
4 ( 7 m ) 3 < / s p an >< / p >< h 2 >< s t ro n g > Q : Wha t i s t h er u l e f or n e g a t i v ee x p o n e n t s ? < / s t ro n g >< / h 2 >< p > A : T h er u l e f or n e g a t i v ee x p o n e n t ss t a t es t ha t < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u p >< mi > a < / mi >< m ro w >< m o > − < / m o >< mi > n < / mi >< / m ro w >< / m s u p >< m o >=< / m o >< m f r a c >< mn > 1 < / mn >< m s u p >< mi > a < / mi >< mi > n < / mi >< / m s u p >< / m f r a c >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > a − n = a n 1 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7713 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > a < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7713 e m ; " >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.1901 e m ; v er t i c a l − a l i g n : − 0.345 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8451 e m ; " >< s p an s t y l e = " t o p : − 2.655 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > a < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.5935 e m ; " >< s p an s t y l e = " t o p : − 2.786 e m ; ma r g in − r i g h t : 0.0714 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.5 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.394 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > . T hi s m e an s t ha t an e g a t i v ee x p o n e n t c anb ere w r i tt e na s a f r a c t i o n w i t h t h e ba se in t h e d e n o mina t or . < / p >< h 2 >< s t ro n g > Q : C an yo ug i v e an e x am pl eo f a ppl y in g t h er u l e f or n e g a t i v ee x p o n e n t s ? < / s t ro n g >< / h 2 >< p > A : L e t ′ ss a y w e w an tt ore w r i t e t h ee x p ress i o n < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m f r a c >< mn > 1 < / mn >< m roo t >< m ro w >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 3 < / mn >< mi > x < / mi >< m s u p >< m os t re t c h y = " f a l se " > ) < / m o >< mn > 5 < / mn >< / m s u p >< / m ro w >< mn > 7 < / mn >< / m roo t >< / m f r a c >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > 7 ( 3 x ) 5 1 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.6747 e m ; v er t i c a l − a l i g n : − 0.8296 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8451 e m ; " >< s p an s t y l e = " t o p : − 2.4642 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d s q r t m t i g h t " >< s p an c l a ss = " roo t " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8307 e m ; " >< s p an s t y l e = " t o p : − 2.8704 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.5 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.0369 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3.4286 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3.4286 e m ; " >< / s p an >< s p an c l a ss = " m or d m t i g h t " s t y l e = " p a dd in g − l e f t : 1.19 e m ; " >< s p an c l a ss = " m o p e nm t i g h t " > ( < / s p an >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > x < / s p an >< s p an c l a ss = " m c l ose m t i g h t " >< s p an c l a ss = " m c l ose m t i g h t " > ) < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7463 e m ; " >< s p an s t y l e = " t o p : − 2.786 e m ; ma r g in − r i g h t : 0.0714 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.5 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 5 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.0089 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3.4286 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l m t i g h t " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.5429 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.5429 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.4197 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.394 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8296 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > u s in g t h er u l e f or n e g a t i v ee x p o n e n t s . W ec an re w r i t e i t a s :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m f r a c >< mn > 1 < / mn >< m roo t >< m ro w >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 3 < / mn >< mi > x < / mi >< m s u p >< m os t re t c h y = " f a l se " > ) < / m o >< mn > 5 < / mn >< / m s u p >< / m ro w >< mn > 7 < / mn >< / m roo t >< / m f r a c >< m o >=< / m o >< m f r a c >< mn > 1 < / mn >< m ro w >< m s u p >< mn > 3 < / mn >< m f r a c >< mn > 5 < / mn >< mn > 7 < / mn >< / m f r a c >< / m s u p >< m s u p >< mi > x < / mi >< m f r a c >< mn > 5 < / mn >< mn > 7 < / mn >< / m f r a c >< / m s u p >< / m ro w >< / m f r a c >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > 7 ( 3 x ) 5 1 = 3 7 5 x 7 5 1 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4514 e m ; v er t i c a l − a l i g n : − 1.13 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.175 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " roo t " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7002 e m ; " >< s p an s t y l e = " t o p : − 2.878 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.5 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.935 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3.2 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3.2 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 1 e m ; " >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 3 < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " >< s p an s t y l e = " t o p : − 2.989 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 5 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.895 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3.2 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 1.02 e m ; h e i g h t : 1.28 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.28 e m " v i e wB o x = "004000001296" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 263 , 681 c 0.7 , 0 , 18 , 39.7 , 52 , 119 c 34 , 79.3 , 68.167 , 158.7 , 102.5 , 238 c 34.3 , 79.3 , 51.8 , 119.3 , 52.5 , 120 c 340 , − 704.7 , 510.7 , − 1060.3 , 512 , − 1067 l 0 − 0 c 4.7 , − 7.3 , 11 , − 11 , 19 , − 11 H 40000 v 40 H 1012.3 s − 271.3 , 567 , − 271.3 , 567 c − 38.7 , 80.7 , − 84 , 175 , − 136 , 283 c − 52 , 108 , − 89.167 , 185.3 , − 111.5 , 232 c − 22.3 , 46.7 , − 33.8 , 70.3 , − 34.5 , 71 c − 4.7 , 4.7 , − 12.3 , 7 , − 23 , 7 s − 12 , − 1 , − 12 , − 1 s − 109 , − 253 , − 109 , − 253 c − 72.7 , − 168 , − 109.3 , − 252 , − 110 , − 252 c − 10.7 , 8 , − 22 , 16.7 , − 34 , 26 c − 22 , 17.3 , − 33.3 , 26 , − 34 , 26 s − 26 , − 26 , − 26 , − 26 s 76 , − 59 , 76 , − 59 s 76 , − 60 , 76 , − 60 z M 100180 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.305 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.13 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.151 e m ; v er t i c a l − a l i g n : − 0.8296 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.1704 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 3 < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " >< s p an s t y l e = " t o p : − 3.3486 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 5 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " >< s p an s t y l e = " t o p : − 3.3486 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 5 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8296 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 2 >< s t ro n g > Q : Ho w d o I s im pl i f y an e x p ress i o n w i t han e g a t i v ee x p o n e n t ? < / s t ro n g >< / h 2 >< p > A : T os im pl i f y an e x p ress i o n w i t han e g a t i v ee x p o n e n t , f o ll o wt h eses t e p s :< / p >< o l >< l i >< s t ro n g > A ppl y t h er u l e f or n e g a t i v ee x p o n e n t s < / s t ro n g >: R e w r i t e t h e n e g a t i v ee x p o n e n t a s a f r a c t i o n w i t h t h e ba se in t h e d e n o mina t or . < / l i >< l i >< s t ro n g > S im pl i f y t h ee x p ress i o n < / s t ro n g >: S im pl i f y t h eres u lt in g e x p ress i o nb yco mbinin g l ik e t er m s . < / l i >< / o l >< h 2 >< s t ro n g > Q : C an yo ug i v e an e x am pl eo f s im pl i f y in g an e x p ress i o n w i t han e g a t i v ee x p o n e n t ? < / s t ro n g >< / h 2 >< p > A : L e t ′ ss a y w e w an tt os im pl i f y t h ee x p ress i o n < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m f r a c >< m ro w >< m s u p >< mn > 3 < / mn >< m ro w >< m o > − < / m o >< m f r a c >< mn > 3 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< m s u p >< mi > x < / mi >< m ro w >< m o > − < / m o >< m f r a c >< mn > 3 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< / m ro w >< m ro w >< m s u p >< mn > 3 < / mn >< m ro w >< m o > − < / m o >< m f r a c >< mn > 2 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< m s u p >< mi > x < / mi >< m ro w >< m o > − < / m o >< m f r a c >< mn > 2 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< / m ro w >< / m f r a c >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > 3 − 7 2 x − 7 2 3 − 7 3 x − 7 3 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.9183 e m ; v er t i c a l − a l i g n : − 0.6671 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.2511 e m ; " >< s p an s t y l e = " t o p : − 2.3329 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.2245 e m ; " >< s p an s t y l e = " t o p : − 3.4878 e m ; ma r g in − r i g h t : 0.0714 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 1 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.0314 e m ; " >< s p an s t y l e = " t o p : − 2.468 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.387 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.532 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 1 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.2245 e m ; " >< s p an s t y l e = " t o p : − 3.4878 e m ; ma r g in − r i g h t : 0.0714 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 1 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.0314 e m ; " >< s p an s t y l e = " t o p : − 2.468 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.387 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.532 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 1 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.394 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.2245 e m ; " >< s p an s t y l e = " t o p : − 3.4878 e m ; ma r g in − r i g h t : 0.0714 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 1 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.0314 e m ; " >< s p an s t y l e = " t o p : − 2.468 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.387 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.532 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 1 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.2245 e m ; " >< s p an s t y l e = " t o p : − 3.4878 e m ; ma r g in − r i g h t : 0.0714 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 1 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.0314 e m ; " >< s p an s t y l e = " t o p : − 2.468 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.387 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.532 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 1 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.6671 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > . F o ll o w in g t h es t e p s ab o v e , w ec an s im pl i f y i t a s :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m f r a c >< m ro w >< m s u p >< mn > 3 < / mn >< m ro w >< m o > − < / m o >< m f r a c >< mn > 3 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< m s u p >< mi > x < / mi >< m ro w >< m o > − < / m o >< m f r a c >< mn > 3 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< / m ro w >< m ro w >< m s u p >< mn > 3 < / mn >< m ro w >< m o > − < / m o >< m f r a c >< mn > 2 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< m s u p >< mi > x < / mi >< m ro w >< m o > − < / m o >< m f r a c >< mn > 2 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< / m ro w >< / m f r a c >< m o >=< / m o >< m f r a c >< m ro w >< m s u p >< mn > 3 < / mn >< m ro w >< m o > − < / m o >< m f r a c >< mn > 3 < / mn >< mn > 7 < / mn >< / m f r a c >< m o > + < / m o >< m f r a c >< mn > 2 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< m s u p >< mi > x < / mi >< m ro w >< m o > − < / m o >< m f r a c >< mn > 3 < / mn >< mn > 7 < / mn >< / m f r a c >< m o > + < / m o >< m f r a c >< mn > 2 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< / m ro w >< m ro w >< m s u p >< mn > 3 < / mn >< m f r a c >< mn > 2 < / mn >< mn > 7 < / mn >< / m f r a c >< / m s u p >< m s u p >< mi > x < / mi >< m f r a c >< mn > 2 < / mn >< mn > 7 < / mn >< / m f r a c >< / m s u p >< / m ro w >< / m f r a c >< m o >=< / m o >< m f r a c >< m ro w >< m s u p >< mn > 3 < / mn >< m ro w >< m o > − < / m o >< m f r a c >< mn > 1 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< m s u p >< mi > x < / mi >< m ro w >< m o > − < / m o >< m f r a c >< mn > 1 < / mn >< mn > 7 < / mn >< / m f r a c >< / m ro w >< / m s u p >< / m ro w >< m ro w >< m s u p >< mn > 3 < / mn >< m f r a c >< mn > 2 < / mn >< mn > 7 < / mn >< / m f r a c >< / m s u p >< m s u p >< mi > x < / mi >< m f r a c >< mn > 2 < / mn >< mn > 7 < / mn >< / m f r a c >< / m s u p >< / m ro w >< / m f r a c >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > 3 − 7 2 x − 7 2 3 − 7 3 x − 7 3 = 3 7 2 x 7 2 3 − 7 3 + 7 2 x − 7 3 + 7 2 = 3 7 2 x 7 2 3 − 7 1 x − 7 1 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4606 e m ; v er t i c a l − a l i g n : − 0.8296 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.631 e m ; " >< s p an s t y l e = " t o p : − 2.1704 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 3 < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " >< s p an s t y l e = " t o p : − 3.3486 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " >< s p an s t y l e = " t o p : − 3.3486 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 3 < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " >< s p an s t y l e = " t o p : − 3.363 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " >< s p an s t y l e = " t o p : − 3.363 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8296 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4606 e m ; v er t i c a l − a l i g n : − 0.8296 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.631 e m ; " >< s p an s t y l e = " t o p : − 2.1704 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 3 < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " >< s p an s t y l e = " t o p : − 3.3486 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " >< s p an s t y l e = " t o p : − 3.3486 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 3 < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " >< s p an s t y l e = " t o p : − 3.363 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< s p an c l a ss = " mbinm t i g h t " > + < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " >< s p an s t y l e = " t o p : − 3.363 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< s p an c l a ss = " mbinm t i g h t " > + < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8296 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4606 e m ; v er t i c a l − a l i g n : − 0.8296 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.631 e m ; " >< s p an s t y l e = " t o p : − 2.1704 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 3 < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " >< s p an s t y l e = " t o p : − 3.3486 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9396 e m ; " >< s p an s t y l e = " t o p : − 3.3486 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 3 < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " >< s p an s t y l e = " t o p : − 3.363 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.954 e m ; " >< s p an s t y l e = " t o p : − 3.363 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : − 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 7 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e m t i g h t " s t y l e = " b or d er − b o tt o m − w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t − s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8296 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 2 >< s t ro n g > Q : Wha t a reso m eco mm o nmi s t ak es t o a v o i d w h e n re w r i t in g e x p o n e n t s in r a d i c a l f or m ? < / s t ro n g >< / h 2 >< p > A : S o m eco mm o nmi s t ak es t o a v o i d w h e n re w r i t in g e x p o n e n t s in r a d i c a l f or min c l u d e :< / p >< u l >< l i >< s t ro n g > N o t f o ll o w in g t h eor d ero f o p er a t i o n s < / s t ro n g >: M ak es u re t o f o ll o wt h eor d ero f o p er a t i o n s ( PEM D A S ) w h e n re w r i t in g e x p o n e n t s in r a d i c a l f or m . < / l i >< l i >< s t ro n g > N o t s im pl i f y in g t h ee x p ress i o n < / s t ro n g >: M ak es u re t os im pl i f y t h eres u lt in g e x p ress i o nb yco mbinin g l ik e t er m s . < / l i >< l i >< s t ro n g > N o t a ppl y in g t h er u l e f or n e g a t i v ee x p o n e n t s < / s t ro n g >: M ak es u re t o a ppl y t h er u l e f or n e g a t i v ee x p o n e n t s w h e n re w r i t in g e x p ress i o n s w i t hn e g a t i v ee x p o n e n t s . < / l i >< / u l >< h 2 >< s t ro n g > Q : Ho w c an I p r a c t i cere w r i t in g e x p o n e n t s in r a d i c a l f or m ? < / s t ro n g >< / h 2 >< p > A : Y o u c an p r a c t i cere w r i t in g e x p o n e n t s in r a d i c a l f or mb y w or kin g t h ro ug h e x erc i ses an d p ro b l e m s t ha t in v o l v ere w r i t in g e x p o n e n t s in r a d i c a l f or m . Y o u c ana l so t ry u s in g o n l in ereso u rcesor ma t h so f tw a re t o h e lp yo u p r a c t i ce an d l e a r n . < / p >