# 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, ${\left(ab\right)}^{m}={a}^{m}\cdot {b}^{m}$.

Example 1:

Simplify ${\left(3t\right)}^{4}$

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

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

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

Example 2:

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

$\begin{array}{l}{3}^{2}\cdot {4}^{2}=\left(3\cdot 3\right)\cdot \left(4\cdot 4\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(3\cdot 4\right)\left(3\cdot 4\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={12}^{2}\end{array}$

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