How to simplify square root

Question

$$ \sqrt[3]{a}(\sqrt[3]{a^2}-\sqrt[3]{a^5}) $$

How can this be simplified? I can't find anything for doing the subtraction.

Answer 1

• Distribute $\;\sqrt[\large 3]{a}\;$ over the difference,
• use the property that $\;\sqrt[\large a]{b}\cdot \sqrt[\large a]{c} = \sqrt[\large a]{b\cdot c},\;$ and
• remember that $\sqrt[\Large a]{b^a} = b$

$$ \begin{align} \sqrt[\Large 3]{a}\left(\sqrt[\Large 3]{a^2}-\sqrt[\Large 3]{a^5}\right) & = \sqrt[\Large 3]{a}\cdot \sqrt[\Large 3]{a^2} - \sqrt[\Large 3]{a}\cdot \sqrt[\Large3]{a^5} \\ \\ & = \sqrt[\Large 3]{a\cdot a^2} - \sqrt[\Large a]{a\cdot a^5} \\ \\ & = \sqrt[\Large 3]{a^3} - \sqrt[\Large 3]{a^6} \\ \\ & = a - \sqrt[\Large 3]{(a^2)^3} \\ \\ & = a \;\;- \;\;a^2 \\ \\ & \end{align} $$

Answer 2

This is $$a^{1/3}(a^{2/3} - a^{5/3}) = a - a^2.$$