8.1: Exponent Properties Involving Products
Learning Objectives
At the end of this lesson, students will be able to:
 Use the product of a power property.
 Use the power of a product property.
 Simplify expressions involving product properties of exponents.
Vocabulary
Terms introduced in this lesson:
 power
 exponent
 square, cube
 base
 factors of the base
 product rule for exponents
 power of a product
 power rule for exponents
Teaching Strategies and Tips
Encourage students to review basic exponents in the chapter Real Numbers.
 An exponent is a notation for repeated multiplication of a number, variable, or expression.
 Exponents count how many bases there are in a product
 Parentheses precede exponents in the order of operations.
Additional Examples:
Write in exponent form.
a. \begin{align*}(4)(4)(4)\end{align*}
b. \begin{align*}(7)(7)(7)(7)\end{align*}
c. \begin{align*}(5w)(5w)(5w)\end{align*}
d. \begin{align*}t \cdot t \cdot t \cdot t \cdot t \cdot t\end{align*}
Encourage proper use of mathematical language:

\begin{align*}3^4\end{align*}
34 is read as “three to the fourth power,” “three to the fourth,” or “three raised to the power of four.” The exponents \begin{align*}2\end{align*}2 and \begin{align*}3\end{align*}3 are special: \begin{align*}3^2\end{align*}32 , “three squared” and \begin{align*}3^3\end{align*}33 , “three cubed.” 
\begin{align*}(3x)^3\end{align*}
(3x)3 is read as “quantity \begin{align*}3x\end{align*}3x cubed.” 
\begin{align*}3^2\end{align*}
−32 is read as “opposite of three squared.” 
\begin{align*}(3)^2\end{align*}
(−3)2 is read as “negative three squared”.
To check for conceptual understanding, ask students to translate the following into symbols.
 Square negative two. Answer: \begin{align*}(2)^2\end{align*}
(−2)2  Negative two squared. Answer: \begin{align*}(2)^2\end{align*}
(−2)2  The opposite of two squared. Answer: \begin{align*}2^2\end{align*}
−22
Allow the class to infer the product rule for exponents in Example 2.
 “When you multiply like bases you add the exponents.”
Allow the class to infer the power rule for exponents in Example 6c.
 “When you raise an exponent to an exponent, you multiply them.”
Combine exponent rules in Examples 69.
 Suggest that students apply each rule one step at a time.
 Order of operations must be followed at each step: evaluate exponents before multiplying. See Examples 6a, 6b, and 9.
Additional Examples:
Simplify the following expressions.
a. \begin{align*}(4x)(3x)^2\end{align*}
Hint: Apply the exponent before multiplying.
b. \begin{align*}(5x^2 + (2x)^2)^3\end{align*}
Hint: Simplify in the parentheses first.
c. \begin{align*}(4x)(3x)^2 + (5x^2 + (2x)^2)^3\end{align*}
Hint: Simplify each term first.
Point out in Example 8 where the commutative property of multiplication is being used.
Error Troubleshooting
Use Examples 4 and 5 to point out two common errors:
 Multiplying bases incorrectly (exponents are different). \begin{align*}2^4 \cdot 2^3 \neq 4^7\end{align*}
24⋅23≠47 .

However, \begin{align*}3^4 \cdot 2^4 = 6^4\end{align*}
34⋅24=64 because \begin{align*}3^4 \cdot 2^4 = (3 \cdot 2)^4 = 6^4\end{align*}34⋅24=(3⋅2)4=64 (exponents are same).
 Applying the product rule incorrectly (bases are different). \begin{align*}3^4 \cdot 2^3 \neq 6^7\end{align*}
34⋅23≠67 .
Remind students in Review Question 6 that even powers of negative numbers are positive. Try using a visual to show that negatives cancel in pairs:
\begin{align*}(2)^6 = (2)(2)(2)(2)(2)(2)=\underbrace{(2)(2)}_{+4} \cdot \underbrace{(2)(2)}_{+4} \cdot \underbrace{(2)(2)}_{+4} = 64\end{align*}
General Tip: Occasionally, students forget that:
 \begin{align*}\mathrm{negative} \times \;\mathrm{positive} = \mathrm{negative}\end{align*}
 \begin{align*}\mathrm{negative} \times \;\mathrm{negative} = \mathrm{positive}\end{align*}
In Review Question 13, remind students that the exponent in \begin{align*}2y^4\end{align*} does not apply to \begin{align*}2\end{align*}. Therefore, \begin{align*}2y^4\end{align*} is simplified.
Additional Examples:
a. Explain the difference between \begin{align*}4x^2\end{align*} and \begin{align*}(4x)^2\end{align*}.
b. Explain the difference between \begin{align*}1^2\end{align*} and \begin{align*}(1)^2\end{align*}.
Teachers are encouraged to survey the class for answers. Ask: What constitutes the base of an exponent?
General Tip: Encourage students to think about syntax before inputting an expression into a calculator.
 \begin{align*}(1)^2\end{align*} often gets inputted incorrectly as \begin{align*}1^{\land}2\end{align*}.
 \begin{align*}\left (\frac{2} {3}\right )^2\end{align*} can get inputted incorrectly as \begin{align*}2/3^{\land}2\end{align*}.