# 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 \(x^{\text-4}\)?

- \(\frac{x^9}{x^5}\)
- \(\frac{x^5}{x^9}\)
- \(\frac{x^{3}}{x^{-1}}\)
- \(x\boldcdot x^{\text-5}\)
- \(\frac{1}{x^4}\)