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.