The Division Rule

The rule for dividing powers is opposite to multiplying. Here is an example to illustrate:

Example 1

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

You split up the power and 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:

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

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

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