Power of a Product

Under What Conditions Do You Use This Rule?

Math

By

Jennifer Ledwith

Updated on October 10, 2018

When to Use the Power of a Product Rule

Definition: (xy)^a = x^ay^b

When this works:

• Condition 1. Two or more variables or constants are being multiplied.

(xy)^a

• Condition 2. The product, or the result of the multiplication, is raised to a power.

(xy)^a

Note: Both conditions must be met.

Use Power of a Product in These Situations:

(2 * 6)⁵
(xy)³
(8x)⁴

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Example: Power of a Product With Constants

Simplify (2 * 6)⁵.

The base is a product of 2 or more constants. Raise each constant by the given exponent.

(2 * 6)⁵= (2)⁵ * (6)⁵

Simplify.

(2)⁵ * (6)⁵= 32 * 7776 = 248,832

Why Does This Work?

Rewrite (2 * 6)⁵

(12)⁵= 12 * 12 * 12 * 12 * 12 = 248,832

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Example: Power of a Product With Variables

Simplify (xy)³

The base is a product of 2 or more variables. Raise each variable by the given exponent.

(x * y)³= x³ * y³ = x³y³

Why Does This Work?

Rewrite (xy)³.

(xy)³= xy * xy * xy = x * x * x * y * y * y

How many x's are there? 3
How many y's are there? 3

Answer: x³y³

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Example: Power of a Product With a Variable and Constant

Simplify (8x)⁴.

The base is a product of a constant and a variable. Raise each by the given exponent.

(8 * x)⁴= (8)⁴ * (x)⁴

Simplify.

(8)⁴ * (x)⁴ = 4,096 * x⁴ = 4,096x⁴

Why Does This Work?

Rewrite (8x)⁴.

(8x)⁴= (8x) * (8x) * (8x) * (8x)

= 8 * 8 * 8 * 8 * x * x * x * x

= 4096x⁴

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Practice Exercises

Check your work with the Answers and Explanations.

Simplify.

1. (ab)⁵

2. (jk)³

3. (8 * 10)²

4. (-3x)⁴

5. (-3x)⁷

6. (abc)¹¹

7. (6pq)⁵

8. (3Π)¹²