Raising an Exponential Expression to a Power
When we raise an exponential expression to a power, we can use the product
rule to find the result, as shown in the following example:
(w^{4 })^{3} 
= w^{4} Â· w^{4} Â· w^{4} 
Three factors of w^{4} because of the exponent 3 

= w^{12} 
Product rule 
By the product rule we add the three 4's to get 12, but 12 is also the
product of 4 and 3.
This example illustrates the power rule for exponents.
Power Rule
If m and n are nonnegative integers and a ≠ 0,
then (a^{ m})^{ n} = a^{ mn}.
Example 1
Using the power rule
Use the rules of exponents to simplify each
expression. Assume that all vairables represent nonzero real numbers.
a) 3x^{2}(x^{3 })^{5}
b)
c)
Solution
a) 3x^{2}(x^{3
})^{5} 
= 3x^{2}x^{15} 
Power rule 

= 3x^{17} 
Product rule 
b)


Power rule and product rule 


Product rule 

= 2^{5} 
Quotient rule 

= 32 
Evaluate 2^{5}. 
c)
