Example with Roots

Here is an example where you use the power rules in combination with $n$th roots.

Example 1

Simplify $\phantom{\rule{-0.17em}{0ex}}{\left(\sqrt[3]{a}×\sqrt[3]{{a}^{2}}\right)}^{2}$

If you use the rules for roots you get that $\sqrt[3]{a}={a}^{\frac{1}{3}}$ and $\sqrt[3]{{a}^{2}}={a}^{\frac{2}{3}}$. This gives you:

 $\phantom{\rule{-0.17em}{0ex}}{\left(\sqrt[3]{a}×\sqrt[3]{{a}^{2}}\right)}^{2}=\phantom{\rule{-0.17em}{0ex}}{\left({a}^{\frac{1}{3}}×{a}^{\frac{2}{3}}\right)}^{2}$

If you now use the rules of calculating with parentheses, you get: $\begin{array}{llll}\hfill \phantom{\rule{-0.17em}{0ex}}{\left({a}^{\frac{1}{3}}×{a}^{\frac{2}{3}}\right)}^{2}& ={a}^{\frac{1×2}{3}}×{a}^{\frac{2×2}{3}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={a}^{\frac{2}{3}+\frac{4}{3}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={a}^{\frac{6}{3}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={a}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$