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 

= 2^{3}x^{3} 
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}
= a^{n }b^{n}.
Example 1
Using the power of a product rule
Simplify. Assume that the variables are nonzero.
a) (xy^{3})^{5}
b) (3m)^{3}
c) (2x^{3}y^{2}z^{7})^{3}
Solution
a) (xy^{3})^{5} 
= x^{5}(y^{3})^{5} 
Power of a product rule 

= x^{5}y^{15} 
Power rule 
b) (3m)^{3} 
= (3)^{3}m^{3} 
Power of a product rule 

= 27m^{3} 
(3)(3)(3) = 27 
c) (2x^{3}y^{2}z^{7})^{3} = 2^{3}(x)^{3}(y^{2})^{3}(z^{7})^{3}
= 8x^{9}y^{6}z^{21}
Power of a Quotient
Raising a quotient to a power is similar to raising a product to a power:

Definition of exponent 3 Definition 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
(5x^{3})^{2} = 5x(x^{3})^{2} = 25x^{6} 
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.
