<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

8.1: Exponent Properties Involving Products

Difficulty Level: At Grade Created by: CK-12

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*}(4)(4)(4)

b. \begin{align*}(-7)(-7)(-7)(-7)\end{align*}(7)(7)(7)(7)

c. \begin{align*}(5w)(5w)(5w)\end{align*}(5w)(5w)(5w)

d. \begin{align*}t \cdot t \cdot t \cdot t \cdot t \cdot t\end{align*}tttttt

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 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*}(4x)(3x)2

Hint: Apply the exponent before multiplying.

b. \begin{align*}(5x^2 + (-2x)^2)^3\end{align*}(5x2+(2x)2)3

Hint: Simplify in the parentheses first.

c. \begin{align*}(4x)(3x)^2 + (5x^2 + (-2x)^2)^3\end{align*}(4x)(3x)2+(5x2+(2x)2)3

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*}242347.
However, \begin{align*}3^4 \cdot 2^4 = 6^4\end{align*}3424=64 because \begin{align*}3^4 \cdot 2^4 = (3 \cdot 2)^4 = 6^4\end{align*}3424=(32)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*}342367.

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*}.

Image Attributions

Show Hide Details
Description
Subjects:
Grades:
Date Created:
Feb 22, 2012
Last Modified:
Aug 22, 2014
Files can only be attached to the latest version of section
Reviews
Help us create better content by rating and reviewing this modality.
Loading reviews...
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
CK.MAT.ENG.TE.1.Algebra-I.8.1

Original text