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8.1: Exponent Properties Involving Products

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. (4)(4)(4)

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

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

d. t \cdot t \cdot t \cdot t \cdot t \cdot t

Encourage proper use of mathematical language:

  • 3^4 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: 3^2, “three squared” and 3^3, “three cubed.”
  • (3x)^3 is read as “quantity 3x cubed.”
  • -3^2 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: -2^2

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. (5x^2 + (-2x)^2)^3

Hint: Simplify in the parentheses first.

c. (4x)(3x)^2 + (5x^2 + (-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). 2^4 \cdot 2^3 \neq 4^7.
However, 3^4 \cdot 2^4 = 6^4 because 3^4 \cdot 2^4 = (3 \cdot 2)^4 = 6^4 (exponents are same).
  • Applying the product rule incorrectly (bases are different). 3^4 \cdot 2^3 \neq 6^7.

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:

(-2)^6 = (-2)(-2)(-2)(-2)(-2)(-2)=\underbrace{(-2)(-2)}_{+4} \cdot \underbrace{(-2)(-2)}_{+4} \cdot \underbrace{(-2)(-2)}_{+4} = 64

General Tip: Occasionally, students forget that:

  • \mathrm{negative} \times \;\mathrm{positive} = \mathrm{negative}
  • \mathrm{negative} \times \;\mathrm{negative} = \mathrm{positive}

In Review Question 13, remind students that the exponent in -2y^4 does not apply to -2. Therefore, -2y^4 is simplified.

Additional Examples:

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

b. Explain the difference between -1^2 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 -1^{\land}2.
  • \left (\frac{2} {3}\right )^2 can get inputted incorrectly as 2/3^{\land}2.

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CK.MAT.ENG.TE.1.Algebra-I.8.1

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