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:

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

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

$$\begin{array}{llll}\hfill {\left(a^{\frac{1}{3}}\times a^{\frac{2}{3}}\right)}^2& =a^{\frac{1\times 2}{3}}\times a^{\frac{2\times 2}{3}}& \hfill & \\ \hfill & =a^{\frac{2}{3}+\frac{4}{3}}& \hfill & \\ \hfill & =a^{\frac{6}{3}}& \hfill & \\ \hfill & =a^2& \hfill & \end{array}$$