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*} 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*} and \begin{align*}3\end{align*} are special: \begin{align*}3^2\end{align*}, “three squared” and \begin{align*}3^3\end{align*}, “three cubed.”
- \begin{align*}(3x)^3\end{align*} is read as “quantity \begin{align*}3x\end{align*} cubed.”
- \begin{align*}-3^2\end{align*} is read as “opposite of three squared.”
- \begin{align*}(-3)^2\end{align*} 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*}
- Negative two squared. Answer: \begin{align*}(-2)^2\end{align*}
- The opposite of two squared. Answer: \begin{align*}-2^2\end{align*}
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 6-9.
- 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*}.
- However, \begin{align*}3^4 \cdot 2^4 = 6^4\end{align*} because \begin{align*}3^4 \cdot 2^4 = (3 \cdot 2)^4 = 6^4\end{align*} (exponents are same).
- Applying the product rule incorrectly (bases are different). \begin{align*}3^4 \cdot 2^3 \neq 6^7\end{align*}.
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*}.