How to Multiply Powers in Parentheses

This entry is a continuation of Rules 1 and 2 about powers, and Rules 3 and 4 about dividing powers.

Rules 5 and 6, presented here, are about powers of parentheses. When you have learned these rules, you should practice examples combining the rules.

Rule 5

Let’s use an example to see why:

$$a^3=a\cdot a\cdot a\text{ and }{b}^3=b\cdot b\cdot b$$

That leads to

$$\begin{array}{llll}\hfill {(ab)}^3& =ab\cdot ab\cdot ab& \hfill & \\ \hfill & =a\cdot b\cdot a\cdot b\cdot a\cdot b& \hfill & \\ \hfill & =a\cdot a\cdot a\cdot b\cdot b\cdot b& \hfill & \\ \hfill & =a^3\cdot b^3.& \hfill & \end{array}$$

Rule 6

Let’s use an example to see why:

$$a^3=a\cdot a\cdot a\text{ and }{b}^3=b\cdot b\cdot b$$

Therefore

$${\left(\frac{a}{b}\right)}^3=\frac{a}{b}\cdot \frac{a}{b}\cdot \frac{a}{b}=\frac{a\cdot a\cdot a}{b\cdot b\cdot b}=\frac{a^3}{b^3}$$