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

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

Summary