Lesson 1

A. \(2^3 = 9\)

B. \(9 = 3^2\)

C. \(2 \boldcdot 2 \boldcdot 2 \boldcdot 2 = 16\)

D. \(a \boldcdot 2^0 = a\)

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. 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 \(7^{\text{-}3}\)?

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.

• \(2^{\text{-}4}\)
• \(\frac{1}{2^4}\)
• \(\text{-}2^4\)
• \(\text{-}\frac{1}{2^4}\)
• \(4^2\)
• \(4^{\text{-}2}\)
• \(\text{-}4^2\)
• \(\text{-}4^{\text{-}2}\)
• \(2^7 \boldcdot 2^{\text{-}3}\)
• \(\frac{2^7}{2^{\text{-}3}}\)
• \(2^{\text{-}7} \boldcdot 2^{3}\)
• \(\frac{2^{\text{-}7}}{2^{\text{-}3}}\)
• \((\text-4)^2\)

 

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.