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Power of a Product Property of Exponents

To find a power of a product, find the power of each factor and then multiply. In general, ${(a b)}^{m} = a^{m} \cdot b^{m}$ .

Example 1:

Simplify ${(3 t)}^{4}$

${(3 t)}^{4} = 3^{4} \cdot t^{4} = 81 t^{4}$

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

$a^{m} \cdot b^{m} = {(a b)}^{m}$ .

Example 2:

Simplify $3^{2} \cdot 4^{2}$

$\begin{array}{l} 3^{2} \cdot 4^{2} = (3 \cdot 3) \cdot (4 \cdot 4) \\ = (3 \cdot 4) (3 \cdot 4) \\ = 12^{2} \end{array}$

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