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I was recently given a problem along the lines of the below:

Simplify $6^{\frac{5}{3}}$ to an expression in the format $a\sqrt[b]{c}$.

The answer, $6\sqrt[3]{36}$, was then given to me before I could figure out the problem myself. I'm wondering what the steps to perform this simplification are, and how they work.

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    $\begingroup$ Or more comically, $a=6^{5/3}$, $b=1$, $c=1$. $\endgroup$
    – parsiad
    Commented Oct 17, 2016 at 17:01
  • $\begingroup$ @parsiad The unwritten request is probably that $a,b,c$ are integers. $\endgroup$
    – Najib Idrissi
    Commented Oct 17, 2016 at 17:08
  • $\begingroup$ One probably also wants $c$ to be positive and cube-free, to ensure uniqueness of the answer. $\endgroup$
    – arkeet
    Commented Oct 17, 2016 at 17:09

3 Answers 3

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\begin{eqnarray*} 6^{5/3} &=& \left(6^5\right)^{1/3} \\ \\ &=& \left(6 \times 6 \times 6 \times 6 \times 6\right)^{1/3} \\ \\ &=& \left(6 \times 6 \times 6\right)^{1/3} \times \left(6 \times 6\right)^{1/3} \\ \\ &=& 6 \times \left(36\right)^{1/3} \\ \\ &=& 6 \times \sqrt[3]{36} \end{eqnarray*}

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$$6^{5/3} = 6^{1 + 2/3} = 6 \cdot 6^{2/3} = 6 \sqrt[3]{6^2}.$$

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$$6^{\frac{5}{3}}=(6^5)^{\frac{1}{3}}=(6^3)^\frac{1}{3}(6^2)^\frac{1}{3}=6(36)^\frac{1}{3}$$

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