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# Power of a Product and Power of a Quotient

## Power of a Product

Consider an example of raising a monomial to a power. We will use known rules to rewrite the expression.

 (2x)3 = 2x Â· 2x Â· 2x Definition of exponent 3 = 2 Â· 2 Â· 2 Â· x Â· x Â· x Commutative and associative properties = 23x3 Definition of exponents

Note that the power 3 is applied to each factor of the product. This example illustrates the power of a product rule.

Power of a Product Rule

If a and b are real numbers and n is a positive integer, then (ab)n = an bn.

Example 1

Using the power of a product rule

Simplify. Assume that the variables are nonzero.

a) (xy3)5

b) (-3m)3

c) (2x3y2z7)3

Solution

 a) (xy3)5 = x5(y3)5 Power of a product rule = x5y15 Power rule
 b) (-3m)3 = (-3)3m3 Power of a product rule = 27m3 (-3)(-3)(-3) = -27

c) (2x3y2z7)3 = 23(x)3(y2)3(z7)3 = 8x9y6z21

## Power of a Quotient

Raising a quotient to a power is similar to raising a product to a power: Definition of exponent 3Definition of multiplication of fractions Definition of exponents

The power is applied to both the numerator and denominator. This example illustrates the power of a quotient rule.

Power of a Quotient Rule

If a and b are real numbers, b ≠ 0, and n is a positive integer, then Example 2

Using the power of a quotient rule

Simplify. Assume that the variables are nonzero. Solution

 a) Power of a quotient rule  (5x3)2 = 5x(x3)2 = 25x6
 b) Power of a quotient and power of a product rules  Simplify

c) Use the quotient rule to simplify the expression inside the parentheses before using the power of a quotient rule.

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