how do you simplify this $\displaystyle {\sqrt[5]{-32}} $?
You're looking for what number raised to the fifth power equals 32.
2x2x2x2x2=32
Since you have an odd numbered root (5th root) the negative sign is ok.
$\displaystyle {\sqrt[5]{-32}}=-2$
I'm not sure if there's something else you're looking for but -2 is as simple as it gets.