Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

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

share|cite|improve this question
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 $\sqrt[\Large a]{b^a} = b$

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

share|cite|improve this answer

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.