# Lesson 7

Negative Exponents

• Let’s explore numbers with negative exponents.

### 7.1: Math Talk: Powers of Ten

Solve each equation mentally:

$$\frac{100}{1}=10^x$$

$$\frac{1000}{x}=10^1$$

$$\frac{x}{100}=10^0$$

$$\frac{100}{1000}=10^x$$

### 7.2: Maintain the Pattern

Complete the table.

exponential form number form calculations
$$2^5$$
16
$$\frac{2^4}{2}=2^{4-1}=2^3$$ $$2^3$$
$$\frac{2^3}{2}=2^{3-1}=2^2$$ $$2^2$$ 4
2 $$4 \boldcdot \frac12=2$$
1 $$2 \boldcdot \frac12=1$$
$$2^{\text-1}$$ $$\frac{1}{2}$$
$$\frac{1}{4}$$ $$\frac12 \boldcdot \frac12 = \frac14$$
$$2^{\text-3}$$
$$2^{\text-4}$$
$$\frac{1}{32}$$

### 7.3: Matching Equal Expressions

Take turns with your partner to match the original expression with an equal or equivalent expression in the list.

• For each match that you find, explain to your partner how you know it’s a match.
• For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

Which expressions equal $$8^0$$?

• 1
• 0
• $$8^3 \boldcdot 8^{-3}$$
• $$\frac{8^2}{8^2}$$
• $$11^0$$

Which expressions equal $$5^{\text-2}$$?

• $$\text-5^2$$
• $$\frac{5^{0}}{5^2}$$
• $$\text-2^5$$
•  $$\frac{1}{5^2}$$
• $$5^{\text-1} \boldcdot 5^{\text-1}$$

Which expressions equal $$3^{10}$$?

• $$3^5\boldcdot3^2$$
•  $$\left(3^5\right)^2$$
•  $$3^7 \boldcdot 3^3$$
•   $$3^{13} \boldcdot 3^{\text-3}$$
•   $$\frac{3^{10}}{3^{0}}$$

Which expressions are equivalent to $$x^{\text-4}$$?

• $$\frac{x^9}{x^5}$$
•  $$\frac{x^5}{x^9}$$
•  $$\frac{x^{3}}{x^{-1}}$$
•   $$x\boldcdot x^{\text-5}$$
•   $$\frac{1}{x^4}$$