How do you evaluate 32 to the power of 2/5? 32^(2/5)

Aug 25, 2016

$4$
There is a choice of methods.. Work smarter, not harder!

Explanation:

One of the laws of indices deals with cases where there are powers and roots at the same time.

${x}^{\frac{p}{q}} = \sqrt[q]{{x}^{p}} = {\left(\sqrt[q]{x}\right)}^{p}$

The denominator shows the root and the numerator gives the power.

Note that the power can be inside or outside the root.

I prefer to find the root first, and then raise to the power because this keeps the numbers smaller. They can usually be calculated mentally rather than needing a calculator

${32}^{\frac{2}{5}} = {\left(\textcolor{red}{\sqrt{32}}\right)}^{2}$

=${\textcolor{red}{2}}^{2} \textcolor{w h i t e}{w w w w w w w w w w w w w} \left(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = {2}^{5} = 32\right)$

=$4$

Compare this with the other method of squaring first.

$\sqrt{\textcolor{b l u e}{{32}^{2}}} = \sqrt{\textcolor{b l u e}{1024}}$

=$4$

While I know that ${2}^{5} = 32$, the square of $32$ and the fifth root of $1024$ are not facts that I would be able to recall from memory.