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8.1: Exponent Properties Involving Products

Difficulty Level: At Grade Created by: CK-12
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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.


Terms introduced in this lesson:

square, cube
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. (4)(4)(4)

b. (7)(7)(7)(7)

c. (5w)(5w)(5w)

d. tttttt

Encourage proper use of mathematical language:

  • 34 is read as “three to the fourth power,” “three to the fourth,” or “three raised to the power of four.” The exponents 2 and 3 are special: 32, “three squared” and 33, “three cubed.”
  • (3x)3 is read as “quantity 3x cubed.”
  • 32 is read as “opposite of three squared.
  • (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: (2)2
  • Negative two squared. Answer: (2)2
  • The opposite of two squared. Answer: 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. (4x)(3x)2

Hint: Apply the exponent before multiplying.

b. (5x2+(2x)2)3

Hint: Simplify in the parentheses first.

c. (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). 242347.
However, 3424=64 because 3424=(32)4=64 (exponents are same).
  • Applying the product rule incorrectly (bases are different). 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:


General Tip: Occasionally, students forget that:

  • negative×positive=negative
  • negative×negative=positive

In Review Question 13, remind students that the exponent in 2y4 does not apply to 2. Therefore, 2y4 is simplified.

Additional Examples:

a. Explain the difference between 4x2 and (4x)2.

b. Explain the difference between 12 and (1)2.

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.

  • (1)2 often gets inputted incorrectly as 12.
  • (23)2 can get inputted incorrectly as 2/32.

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