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Raising an Exponential Expression to a Power

An expression such as (x3)2 consists of the exponential expression x3 raised to the power 2. We can use known rules to simplify this expression.

(x3)2 = x3 · x3 Exponent 2 indicates two factors of x3.
  = x6 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 (am)n = amn.

 

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) (23)5 = 215 Power of a power rule
b) (x2)-6 = x-12 Power of a power rule
  Definition of a negative exponent
c) 3(y-3)-2y-5 = 3y6y-5 Power of a power rule
  = 3y Product rule
d) Power of a power rule
  = x7 Quotient rule

 

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