Raising an Exponential Expression to a Power
An expression such as (x^{3})^{2} consists of the exponential expression x^{3} raised to the
power 2. We can use known rules to simplify this expression.
(x^{3})^{2} 
= x^{3} Â· x^{3} 
Exponent 2 indicates two factors of x^{3}. 

= x^{6} 
Product rule: 3 + 3 = 6 
Note that the exponent 6 is the product of the exponents 2 and 3. This example
illustrates the power of a power rule.
Power of a Power Rule
If m and n are any integers and a ≠ 0, then
(a^{m})^{n} = a^{mn}.
Example
Using the power of a power rule
Use the rules of exponents to simplify each expression. Write the answer with
positive exponents only. Assume all variables represent nonzero real numbers.
Solution
a) (2^{3})^{5} = 2^{15} 
Power of a power rule 
b) (x^{2})^{6} 
= x^{12} 
Power of a power rule 


Definition of a negative exponent 
c) 3(y^{3})^{2}y^{5} 
= 3y^{6}y^{5} 
Power of a power rule 

= 3y 
Product rule 
d)


Power of a power rule 

= x^{7} 
Quotient rule 
