How do you solve 32 to the power of 2/5?

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Explanation

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Explanation:

I want someone to double check my answer

35
Nov 6, 2015

Explanation:

$x = {\left(32\right)}^{\frac{2}{5}}$ can be solved by using the exponent rule for fractional exponents: ${x}^{\frac{m}{n}} = \sqrt[n]{{x}^{m}}$. So, we can rewrite ${\left(32\right)}^{\frac{2}{5}}$ as $\sqrt[5]{{32}^{2}}$

${32}^{2}$ evaluates to 1024, and $\sqrt[5]{1024}$, or the 5th root of 1024 (what times itself 5 times equals 1024) equals 4.

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15
Sep 16, 2017

$4$
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.

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

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

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

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

=$4$

Compare this with the other method of squaring first.

$\sqrt[5]{\textcolor{b l u e}{{32}^{2}}} = \sqrt[5]{\textcolor{b l u e}{1024}}$

=$4$

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.

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