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

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

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2 Answers

up vote 6 down vote accepted
  • 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 $\sqrt[\Large a]{b^a} = b$


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

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

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