# Lesson 1

Properties of Exponents

• Let’s use integer exponents.

### 1.1: Which One Doesn’t Belong: Exponents and Equations

A. $$2^3 = 9$$

B. $$9 = 3^2$$

C. $$2 \boldcdot 2 \boldcdot 2 \boldcdot 2 = 16$$

D. $$a \boldcdot 2^0 = a$$

### 1.2: Name That Power

Find the value of each variable that makes the equation true. Be prepared to explain your reasoning.

1. $$2^3 \boldcdot 2^5 = 2^a$$
2. $$3^b \boldcdot 3^7 = 3^{11}$$
3. $$\frac{4^3}{4^2} = 4^c$$
4. $$\frac{5^8}{5^d} = 5^2$$
5. $$6^m \boldcdot 6^m \boldcdot 6^m = 6^{21}$$
6. $$(7^n)^4 = 7^{20}$$
7. $$2^4 \boldcdot 3^4 = 6^s$$
8. $$5^3 \boldcdot t^3 = 50^3$$

### 1.3: The Power of Zero

1. Use exponent rules to write each expression as a single power of 2. Find the value of the expression. Record these in the table. The first row is done for you.
expression power of 2 value
$$\frac{2^5}{2^1}$$ $$2^4$$ 16
$$\frac{2^5}{2^2}$$
$$\frac{2^5}{2^3}$$
$$\frac{2^5}{2^4}$$
$$\frac{2^5}{2^5}$$
$$\frac{2^5}{2^6}$$
$$\frac{2^5}{2^7}$$
2. What is the value of $$5^0$$?
3. What is the value of $$3^{\text{-}1}$$?
4. What is the value of $$7^{\text{-}3}$$?

Explain why the argument used to assign a value to the expression $$2^0$$ does not apply to make sense of the expression $$0^0$$.

### 1.4: Matching Exponent Expressions

Sort expressions that are equal into groups. Some expressions may not have a match, and some may have more than one match. Be prepared to explain your reasoning.

• $$2^{\text{-}4}$$
• $$\frac{1}{2^4}$$
• $$\text{-}2^4$$
• $$\text{-}\frac{1}{2^4}$$
• $$4^2$$
• $$4^{\text{-}2}$$
• $$\text{-}4^2$$
• $$\text{-}4^{\text{-}2}$$
• $$2^7 \boldcdot 2^{\text{-}3}$$
• $$\frac{2^7}{2^{\text{-}3}}$$
• $$2^{\text{-}7} \boldcdot 2^{3}$$
• $$\frac{2^{\text{-}7}}{2^{\text{-}3}}$$
• $$(\text-4)^2$$

### Summary

Exponent rules help us keep track of a base’s repeated factors. Negative exponents help us keep track of repeated factors that are the reciprocal of the base. We can define a number to the power of 0 to have a value of 1. These rules can be written symbolically as:

\begin{align}b^m \boldcdot b^n &= b^{m+n} \\ \left(b^m\right)^n &= b^{m \boldcdot n} \\ \frac{b^m}{b^n} &= b^{m-n} \\ b^{\text-n} &= \frac{1}{b^n} \\ b^0 &= 1 \\ a^n \boldcdot b^n &= (a \boldcdot b)^n \end{align}

Here, the base $$b$$ can be any positive number, and the exponents $$n$$ and $$m$$ can be any integer.