The Product and Quotient Rules
The rules that we have already discussed are summarized below.
The following rules hold for nonnegative integers m and n and a
≠ 0.
a^{ m} Â· a^{ n} = a^{ m + n} 
Product rule 

Quotient rule 

a^{ 0} = 1 
Zero exponent 
Cuation
The product and quotient rules apply only if the bases of the expressions are
identical. For example, 3^{2} Â· 3^{4} = 3^{6},
but the product rule cannot be applied to 5^{2} Â· 3^{4}. Note
also that the bases are not multiplied: 3^{2} Â· 3^{4}
≠ 9^{6}.
Note that in the quotient rule the exponents are
always subtracted, as in
If the larger exponent is in the denominator,
then the result is placed in the denominator.
Example 1
Using the product and quotient rules
Use the rules of exponents to simplify each
expression. Assume that all variables represent nonzero real numbers.
Solution
a) Because the bases are both 2, we can use the
product rule:
2^{3} Â· 2^{2} 
= 2^{5} 
Product rule 

= 32 
Simplify 
b) (3x)^{0}(5x^{2})(4x) 
= 1 Â· 5x^{2} Â· 4x 
Definition of zero exponent 

= 20x^{3} 
Product rule 
c)

Quotient rule 
d) First use the product rule to simplify the numerator and denominator:

Product rule 
Quotient rule 
