Example with Roots

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

Example 1

Simplify ${(\sqrt[3]{a} \times \sqrt[3]{a^{2}})}^{2}$

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:

{(\sqrt[3]{a} \times \sqrt[3]{a^{2}})}^{2} = {(a^{\frac{1}{3}} \times a^{\frac{2}{3}})}^{2}

If you now use the rules of calculating with parentheses, you get:

\begin{array}{l} {(a^{\frac{1}{3}} \times a^{\frac{2}{3}})}^{2} & = a^{\frac{1 \times 2}{3}} \times a^{\frac{2 \times 2}{3}} \\ = a^{\frac{2}{3} + \frac{4}{3}} \\ = a^{\frac{6}{3}} \\ = a^{2} \end{array}

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