# The Multiplication Rule

The rule for multiplication with powers is simple: If the two powers have the same base, you add their exponents together. It can best be illustrated by examples.

Example 1

Calculate ${2}^{3}×{2}^{2}$

If you use the definition of powers, you see that this can be split up like this:

 $\underset{{2}^{3}}{\underbrace{\phantom{\rule{4.0pt}{0ex}}2×2×2\phantom{\rule{4.0pt}{0ex}}}}×\underset{{2}^{2}}{\underbrace{\phantom{\rule{3.0pt}{0ex}}2×2\phantom{\rule{3.0pt}{0ex}}}}={2}^{5}$

You see from this example that all you need to do is to add the exponents:

 ${2}^{3}×{2}^{2}={2}^{3+2}={2}^{5}$

Rule

### MultiplicationofPowerswiththeSameBase

 ${a}^{n}×{a}^{m}={a}^{n+m}$

Example 2

Calculate ${2}^{3}×{2}^{-2}$

Using the rule above and that $+\left(-\right)=-$, you get:

 ${2}^{3}×{2}^{-2}={2}^{3+\left(-2\right)}={2}^{3-2}={2}^{1}=2$