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# 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.

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.

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.

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 $2/3^{\land}2$.

Feb 22, 2012

Aug 22, 2014