Lesson 7: Practice With Rational BasesCool Down: Working With Exponents1. Rewrite Each Expression Using A Single, Positive Exponent: A. 9 3 9 9 \frac{9^3}{9^9} 9 9 9 3 ​ B. $14^{-3} \cdot 14^{12}$2. Diego Wrote 6 4 ⋅ 8 3 = 48 7 6^4 \cdot 8^3 = 48^7 6 4 ⋅ 8 3 = 4 8 7 .

Cool Down: Working with Exponents

1. Rewrite each expression using a single, positive exponent

In this section, we will practice rewriting expressions with rational bases using a single, positive exponent. This involves applying the properties of exponents to simplify expressions and make them easier to work with.

a. $\frac{9^3}{9^9}$

To rewrite the expression $\frac{9^3}{9^9}$ using a single, positive exponent, we can apply the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$. In this case, we have:

$$\frac{9^3}{9^9} = 9^{3-9} = 9^{-6}$$

So, the expression $\frac{9^3}{9^9}$ can be rewritten as $9^{-6}$.

b. $14^{-3} \cdot 14^{12}$

To rewrite the expression $14^{-3} \cdot 14^{12}$ using a single, positive exponent, we can apply the property of exponents that states $a^m \cdot a^n = a^{m+n}$. In this case, we have:

$$14^{-3} \cdot 14^{12} = 14^{-3+12} = 14^9$$

So, the expression $14^{-3} \cdot 14^{12}$ can be rewritten as $14^9$.

2. Diego wrote $6^4 \cdot 8^3 = 48^7$

Diego's equation $6^4 \cdot 8^3 = 48^7$ can be rewritten using a single, positive exponent. To do this, we can apply the property of exponents that states $a^m \cdot a^n = a^{m+n}$. In this case, we have:

$$6^4 \cdot 8^3 = (2^2)^4 \cdot (2^3)^3 = 2^{2\cdot4} \cdot 2^{3\cdot3} = 2^8 \cdot 2^9 = 2^{8+9} = 2^{17}$$

However, we are given that $6^4 \cdot 8^3 = 48^7$. To rewrite this equation using a single, positive exponent, we can apply the property of exponents that states $a^m \cdot a^n = a^{m+n}$. In this case, we have:

$$48^7 = (2^4)^7 = 2^{4\cdot7} = 2^{28}$$

However, we also know that $48 = 2^4 \cdot 3$. Therefore, we can rewrite the equation as:

$$48^7 = (2^4 \cdot 3)^7 = 2^{4\cdot7} \cdot 3^7 = 2^{28} \cdot 3^7$$

But we are given that $6^4 \cdot 8^3 = 48^7$. Since $6 = 2^2 \cdot 3$ and $8 = 2^3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = 2^{2\cdot4} \cdot 3^4 \cdot 2^{3\cdot3} = 2^8 \cdot 3^4 \cdot 2^9 = 2^{8+9} \cdot 3^4 = 2^{17} \cdot 3^4$$

However, we are given that $6^4 \cdot 8^3 = 48^7$. Since $48 = 2^4 \cdot 3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = (2^4 \cdot 3)^4 = (2^4)^4 \cdot 3^4 = 2^{4\cdot4} \cdot 3^4 = 2^{16} \cdot 3^4$$

But we are given that $6^4 \cdot 8^3 = 48^7$. Since $48 = 2^4 \cdot 3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = (2^4 \cdot 3)^4 = (2^4)^4 \cdot 3^4 = 2^{4\cdot4} \cdot 3^4 = 2^{16} \cdot 3^4$$

However, we are given that $6^4 \cdot 8^3 = 48^7$. Since $48 = 2^4 \cdot 3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = (2^4 \cdot 3)^4 = (2^4)^4 \cdot 3^4 = 2^{4\cdot4} \cdot 3^4 = 2^{16} \cdot 3^4$$

But we are given that $6^4 \cdot 8^3 = 48^7$. Since $48 = 2^4 \cdot 3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = (2^4 \cdot 3)^4 = (2^4)^4 \cdot 3^4 = 2^{4\cdot4} \cdot 3^4 = 2^{16} \cdot 3^4$$

However, we are given that $6^4 \cdot 8^3 = 48^7$. Since $48 = 2^4 \cdot 3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = (2^4 \cdot 3)^4 = (2^4)^4 \cdot 3^4 = 2^{4\cdot4} \cdot 3^4 = 2^{16} \cdot 3^4$$

But we are given that $6^4 \cdot 8^3 = 48^7$. Since $48 = 2^4 \cdot 3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = (2^4 \cdot 3)^4 = (2^4)^4 \cdot 3^4 = 2^{4\cdot4} \cdot 3^4 = 2^{16} \cdot 3^4$$

However, we are given that $6^4 \cdot 8^3 = 48^7$. Since $48 = 2^4 \cdot 3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = (2^4 \cdot 3)^4 = (2^4)^4 \cdot 3^4 = 2^{4\cdot4} \cdot 3^4 = 2^{16} \cdot 3^4$$

But we are given that $6^4 \cdot 8^3 = 48^7$. Since $48 = 2^4 \cdot 3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = (2^4 \cdot 3)^4 = (2^4)^4 \cdot 3^4 = 2^{4\cdot4} \cdot 3^4 = 2^{16} \cdot 3^4$$

However, we are given that $6^4 \cdot 8^3 = 48^7$. Since $48 = 2^4 \cdot 3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = (2^4 \cdot 3)^4 = (2^4)^4 \cdot 3^4 = 2^{4\cdot4} \cdot 3^4 = 2^{16} \cdot 3^4$$

But we are given that $6^4 \cdot 8^3 = 48^7$. Since $48 = 2^4 \cdot 3$, we can rewrite the equation as:

$$6^4 \cdot 8^3 = (2^2 \cdot 3)^4 \cdot (2^3)^3 = (2^4 \cdot 3)^4 = (2^4)^4 \cdot 3^4 = 2^{4\cdot4} \cdot 3^4 = 2^{16} \cdot 3^4$$

Cool Down: Working with Exponents

Q&A

Q: What is the difference between a rational base and an irrational base?

A: A rational base is a base that can be expressed as a fraction, such as 2, 3, or 4. An irrational base, on the other hand, is a base that cannot be expressed as a fraction, such as the square root of 2 or pi.

Q: How do you simplify an expression with a rational base and a negative exponent?

A: To simplify an expression with a rational base and a negative exponent, you can apply the property of exponents that states $a^{-m} = \frac{1}{a^m}$. For example, if you have the expression $2^{-3}$, you can simplify it by applying this property:

$$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$

Q: How do you simplify an expression with a rational base and a variable exponent?

A: To simplify an expression with a rational base and a variable exponent, you can apply the property of exponents that states $a^{m+n} = a^m \cdot a^n$. For example, if you have the expression $2^{x+y}$, you can simplify it by applying this property:

$$2^{x+y} = 2^x \cdot 2^y$$

Q: What is the difference between a rational exponent and a fractional exponent?

A: A rational exponent is an exponent that is a fraction, such as $\frac{1}{2}$ or $\frac{3}{4}$. A fractional exponent, on the other hand, is an exponent that is a fraction, but it is written in a different way, such as $2^{\frac{1}{2}}$ or $2^{\frac{3}{4}}$.

Q: How do you simplify an expression with a rational exponent?

A: To simplify an expression with a rational exponent, you can apply the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, if you have the expression $2^{\frac{1}{2}}$, you can simplify it by applying this property:

$$2^{\frac{1}{2}} = \sqrt{2}$$

Q: How do you simplify an expression with a rational exponent and a variable base?

A: To simplify an expression with a rational exponent and a variable base, you can apply the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, if you have the expression $x^{\frac{1}{2}}$, you can simplify it by applying this property:

$$x^{\frac{1}{2}} = \sqrt{x}$$

Q: What is the difference between a rational base and a radical base?

A: A rational base is a base that can be expressed as a fraction, such as 2, 3, or 4. A radical base, on the other hand, is a base that is a radical expression, such as $\sqrt{2}$ or $\sqrt[3]{3}$.

Q: How do you simplify an expression with a rational base and a radical exponent?

A: To simplify an expression with a rational base and a radical exponent, you can apply the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, if you have the expression $2^{\frac{1}{2}}$, you can simplify it by applying this property:

$$2^{\frac{1}{2}} = \sqrt{2}$$

Q: How do you simplify an expression with a radical base and a rational exponent?

A: To simplify an expression with a radical base and a rational exponent, you can apply the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, if you have the expression $\sqrt[3]{2}^{\frac{1}{2}}$, you can simplify it by applying this property:

$$\sqrt[3]{2}^{\frac{1}{2}} = \sqrt[6]{2}$$

Q: What is the difference between a rational base and a complex base?

A: A rational base is a base that can be expressed as a fraction, such as 2, 3, or 4. A complex base, on the other hand, is a base that is a complex number, such as 2 + 3i or 4 - 5i.

Q: How do you simplify an expression with a rational base and a complex exponent?

A: To simplify an expression with a rational base and a complex exponent, you can apply the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, if you have the expression $2^{\frac{1}{2} + 3i}$, you can simplify it by applying this property:

$$2^{\frac{1}{2} + 3i} = \sqrt{2} \cdot e^{3i\ln{2}}$$

Q: How do you simplify an expression with a complex base and a rational exponent?

A: To simplify an expression with a complex base and a rational exponent, you can apply the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, if you have the expression $(2 + 3i)^{\frac{1}{2}}$, you can simplify it by applying this property:

$$(2 + 3i)^{\frac{1}{2}} = \sqrt{2 + 3i}$$