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Simplifying Exponents - Power of a Product

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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

1. 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

 

2. 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

 

3. 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

 

4. Practice Exercises

Check your work with the Answers and Explanations. (Link Here)

 

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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