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# 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:

 (w4 )3 = w4 Â· w4 Â· w4 Three factors of w4 because of the exponent 3 = w12 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) 3x2(x3 )5

b)

c)

Solution

 a) 3x2(x3 )5 = 3x2x15 Power rule = 3x17 Product rule
 b) Power rule and product rule Product rule = 25 Quotient rule = 32 Evaluate 25.

c)