Select All The Expressions That Are Equivalent To 12 2 3 2 \frac{12^2}{3^2} 3 2 1 2 2 ​ .A. 1 4 2 \frac{1}{4^2} 4 2 1 ​ B. 4 C. 4 2 4^2 4 2 D. ( 4 1 ) 2 \left(4^1\right)^2 ( 4 1 ) 2

Introduction

In mathematics, simplifying exponential expressions is a crucial skill that helps us solve complex problems with ease. One such expression is $\frac{12^2}{3^2}$. In this article, we will explore the different ways to simplify this expression and identify the equivalent expressions from the given options.

Understanding Exponential Expressions

Before we dive into simplifying the given expression, let's understand what exponential expressions are. An exponential expression is a mathematical expression that involves a base raised to a power. For example, $2^3$ is an exponential expression where 2 is the base and 3 is the exponent.

Simplifying the Given Expression

Now, let's simplify the given expression $\frac{12^2}{3^2}$. To simplify this expression, we can use the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$.

Using this rule, we can rewrite the given expression as:

$$\frac{12^2}{3^2} = \frac{(3 \cdot 4)^2}{3^2}$$

Now, we can simplify the expression further by canceling out the common factor of $3^2$ in the numerator and denominator:

$$\frac{(3 \cdot 4)^2}{3^2} = \frac{4^2 \cdot 3^2}{3^2} = 4^2$$

Analyzing the Options

Now that we have simplified the given expression, let's analyze the options:

A. $\frac{1}{4^2}$

B. 4

C. $4^2$

D. $\left(4^1\right)^2$

From our simplification, we can see that the correct answer is C. $4^2$. This is because we simplified the given expression to $4^2$, which is equivalent to option C.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill that helps us solve complex problems with ease. By understanding the rules of exponents and applying them to the given expression, we can simplify it to $4^2$. This is equivalent to option C, which is the correct answer.

Tips and Tricks

Here are some tips and tricks to help you simplify exponential expressions like the one we discussed:

• Use the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify expressions.
• Cancel out common factors in the numerator and denominator to simplify expressions.
• Use the property of exponents that states $(a^m)^n = a^{m \cdot n}$ to simplify expressions.

By following these tips and tricks, you can simplify exponential expressions like the one we discussed and solve complex problems with ease.

Frequently Asked Questions

Here are some frequently asked questions about simplifying exponential expressions:

Q: What is the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$? A: This rule states that when we divide two exponential expressions with the same base, we can subtract the exponents.

Q: How do I simplify an expression like $\frac{12^2}{3^2}$? A: To simplify this expression, we can use the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$. We can also cancel out common factors in the numerator and denominator to simplify the expression.

Q: What is the property of exponents that states $(a^m)^n = a^{m \cdot n}$? A: This property states that when we raise an exponential expression to a power, we can multiply the exponents.

By understanding these rules and properties, you can simplify exponential expressions like the one we discussed and solve complex problems with ease.

References

Here are some references that you can use to learn more about simplifying exponential expressions:

• Math Is Fun
• Khan Academy
• Purplemath

Introduction

In our previous article, we discussed how to simplify exponential expressions using the rules of exponents. In this article, we will provide a Q&A guide to help you understand and apply these rules to simplify exponential expressions.

Q&A Guide

Q: What is the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$? A: This rule states that when we divide two exponential expressions with the same base, we can subtract the exponents. For example, $\frac{2^3}{2^2} = 2^{3-2} = 2^1 = 2$.

Q: How do I simplify an expression like $\frac{12^2}{3^2}$? A: To simplify this expression, we can use the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$. We can also cancel out common factors in the numerator and denominator to simplify the expression. In this case, we can rewrite the expression as $\frac{(3 \cdot 4)^2}{3^2}$ and then simplify it to $4^2$.

Q: What is the property of exponents that states $(a^m)^n = a^{m \cdot n}$? A: This property states that when we raise an exponential expression to a power, we can multiply the exponents. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

Q: How do I simplify an expression like $2^{3 \cdot 4}$? A: To simplify this expression, we can use the property of exponents that states $(a^m)^n = a^{m \cdot n}$. In this case, we can rewrite the expression as $2^{12}$.

Q: What is the rule of exponents that states $a^m \cdot a^n = a^{m+n}$? A: This rule states that when we multiply two exponential expressions with the same base, we can add the exponents. For example, $2^3 \cdot 2^2 = 2^{3+2} = 2^5$.

Q: How do I simplify an expression like $2^3 \cdot 2^2$? A: To simplify this expression, we can use the rule of exponents that states $a^m \cdot a^n = a^{m+n}$. In this case, we can rewrite the expression as $2^5$.

Q: What is the property of exponents that states $a^{-m} = \frac{1}{a^m}$? A: This property states that when we have a negative exponent, we can rewrite it as a fraction with the base in the denominator. For example, $2^{-3} = \frac{1}{2^3}$.

Q: How do I simplify an expression like $2^{-3}$? A: To simplify this expression, we can use the property of exponents that states $a^{-m} = \frac{1}{a^m}$. In this case, we can rewrite the expression as $\frac{1}{2^3}$.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill that helps us solve complex problems with ease. By understanding the rules of exponents and applying them to the given expressions, we can simplify them and solve problems with confidence.

Tips and Tricks

Here are some tips and tricks to help you simplify exponential expressions like the ones we discussed:

• Use the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify expressions.
• Use the property of exponents that states $(a^m)^n = a^{m \cdot n}$ to simplify expressions.
• Use the rule of exponents that states $a^m \cdot a^n = a^{m+n}$ to simplify expressions.
• Use the property of exponents that states $a^{-m} = \frac{1}{a^m}$ to simplify expressions.

By following these tips and tricks, you can simplify exponential expressions like the ones we discussed and solve complex problems with ease.

Frequently Asked Questions

Here are some frequently asked questions about simplifying exponential expressions:

Q: What is the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$? A: This rule states that when we divide two exponential expressions with the same base, we can subtract the exponents.

Q: How do I simplify an expression like $\frac{12^2}{3^2}$? A: To simplify this expression, we can use the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$. We can also cancel out common factors in the numerator and denominator to simplify the expression.

Q: What is the property of exponents that states $(a^m)^n = a^{m \cdot n}$? A: This property states that when we raise an exponential expression to a power, we can multiply the exponents.

By following these FAQs, you can understand and apply the rules of exponents to simplify exponential expressions like the ones we discussed.

References

Here are some references that you can use to learn more about simplifying exponential expressions:

• Math Is Fun
• Khan Academy
• Purplemath

By following these references, you can learn more about simplifying exponential expressions and improve your math skills.