This entry is a continuation of Rules 1 and 2 about powers. Rules 3 and 4, presented here, tell you how to divide powers. The next entry will present Rules 5 and 6 about powers of parentheses.

When you have learned these rules, you should practice examples combining the rules.

Rule 3

Let’s use an example to see why:

$$a^5=a\cdot a\cdot a\cdot a\cdot a$$

and

$$a^3=a\cdot a\cdot a$$

This means that

$$\begin{array}{llll}\hfill a^5÷a^3& =(a\cdot a\cdot a\cdot a\cdot a)÷(a\cdot a\cdot a)& \hfill & \\ \hfill & =\frac{a\cdot a\cdot a\cdot a\cdot a}{a\cdot a\cdot a}& \hfill & \\ \hfill & =\frac{\text{<mi>a</mi>}\cdot \text{<mi>a</mi>}\cdot \text{<mi>a</mi>}\cdot a\cdot a}{\text{<mi>a</mi>}\cdot \text{<mi>a</mi>}\cdot \text{<mi>a</mi>}}& \hfill & \\ \hfill & =a^2& \hfill & \end{array}$$

Rule 4

Let’s apply Rule 3 in an example to see why Rule 4 is true:

$$a^{-n}=a^{0-n}=a^0÷a^n=\frac{a^0}{a^n}=\frac{1}{a^n}$$