How do you simplify the expression (1/32)^(-2/5)?

Apr 20, 2016

${\left(\frac{1}{32}\right)}^{- \frac{2}{5}} = 4$

Explanation:

To make this easier to solve, there's a rule that helps: ${a}^{m n} = {\left({a}^{m}\right)}^{n}$, and what it basically says is that you can split up to the index/exponent (the small raised number) into smaller numbers which multiply to it, e.g. ${2}^{6} = {2}^{2 \cdot 3} = {\left({2}^{2}\right)}^{3}$ or ${2}^{27} = {2}^{3 \cdot 3 \cdot 3} = {\left({\left({2}^{3}\right)}^{3}\right)}^{3}$

Ok let's make that number less scary by spreading it out:
${\left(\frac{1}{32}\right)}^{- \frac{2}{5}} = {\left({\left({\left(\frac{1}{32}\right)}^{-} 1\right)}^{\frac{1}{5}}\right)}^{2}$
Now lets solve from the inside out.
$= {\left({\left(32\right)}^{\frac{1}{5}}\right)}^{2}$
We can say this because: ${\left(\frac{1}{32}\right)}^{-} 1 = \frac{32}{1} = 32$, and then we replace it within the equation. *Note: a '-1' exponent means to just flip the fraction or number*

$= {\left(2\right)}^{2}$

We can say this because ${32}^{\frac{1}{5}} = 2$ *Note: Unless you know logarithms, there's no way to know this other than using your calculator. Also, if the exponent is a fraction, it means to 'root' it e.g. ${8}^{\frac{1}{3}} = \sqrt{2}$*

$= 4$

Last and easy step