Power of a product rule for exponents

The product rule lets us multiply exponents more easily. It states the product of two exponent terms with the same base is the base raised to the sum of the exponents.

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To find a power of a product, find the power of each factor and then multiply.  In general, ( ab ) m = a m ⋅ b m .

Example 1:

Simplify ( 3t ) 4

( 3t ) 4 = 3 4 ⋅ t 4 =81 t 4

Suppose you want to multiply two powers with the same exponent but different bases.  By using the commutative property of multiplication, you can rewrite the rule as

          a m ⋅ b m = ( ab ) m .

Example 2:

Simplify 3 2 ⋅ 4 2

3 2 ⋅ 4 2 =( 3⋅3 )⋅( 4⋅4 )              =( 3⋅4 )( 3⋅4 )              = 12 2

In other words, you can keep the exponent the same and multiply the bases.