Power of a Product

Under What Conditions Do You Use This Rule?

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By
Jennifer Ledwith
Jennifer Ledwith
Math Expert
  • B.B.A., Finance and Economics, University of Oklahoma
Jennifer Ledwith is the owner of tutoring and test-preparation company Scholar Ready, LLC and a professional writer, covering math-related topics.
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Updated on October 10, 2018

When to Use the Power of a Product Rule

Definition:  (xy)a = xayb

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
01
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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

Exponential curve
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Exponents and Bases
By Jennifer Ledwith
02
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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

03
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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

04
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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

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Your Citation
Ledwith, Jennifer. "Power of a Product." ThoughtCo, Apr. 5, 2023, thoughtco.com/power-of-a-product-2312221. Ledwith, Jennifer. (2023, April 5). Power of a Product. Retrieved from https://www.thoughtco.com/power-of-a-product-2312221 Ledwith, Jennifer. "Power of a Product." ThoughtCo. https://www.thoughtco.com/power-of-a-product-2312221 (accessed April 27, 2024).