# The Division Rule

The rule for dividing powers is the opposite of multiplying: If the two powers have the same base, you subtract their exponents from each other. Here is an example to illustrate:

Example 1

Calculate $\frac{{2}^{3}}{{2}^{2}}$

You split up the power, cancel equal factors, and get:

 $\frac{{2}^{3}}{{2}^{2}}=\frac{2×2×2}{2×2}=\frac{\text{2}×\text{2}×2}{\text{2}×\text{2}}=2$

You see that what has happened is that you have subtracted the exponents:

 $\frac{{2}^{3}}{{2}^{2}}={2}^{3-2}={2}^{1}=2$

Rule

### DivisionofPowerswiththeSameBase

 $\frac{{a}^{n}}{{a}^{m}}={a}^{n-m}$

Example 2

Calculate $\frac{{2}^{5}}{{2}^{-2}}$

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

 $\frac{{2}^{5}}{{2}^{-2}}={2}^{5-\left(-2\right)}={2}^{5+2}={2}^{7}$