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