## When to Use the Power of a Product Rule

**Definition**: (*xy*)^{a} = *x ^{a}y^{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} - (8
*x*)^{4}

## 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**

## 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 }= *x*^{3} * *y*^{3} =** x^{3}y^{3}**

## 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: *x*^{3}*y*^{3}

## Example: Power of a Product With a Variable and Constant

Simplify (8*x*)^{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 * *x*^{4} = **4,096 x^{4}**

## Why Does This Work?

Rewrite (8*x*)^{4}.

(8*x*)^{4 }= (8x) * (8x) * (8x) * (8x)

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

= 4096*x*^{4}

## Practice Exercises

Check your work with the Answers and Explanations.

Simplify.

1. (*ab*)^{5}

2. (*jk*)^{3}

3. (8 * 10)^{2}

4. (-3*x*)^{4}

5. (-3*x*)^{7}

6. (*abc*)^{11}

7. (6*pq*)^{5}

8. (3*Π*)^{12}