TITLE: How do you compute negative numbers to fractional powers?
QUESTION [70 upvotes]: My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure if the number stays negative or turns into a positive. I know that if a negative number is raised to an odd power it is negative, but fractional powers are neither odd or even. Is there a general rule for dealing with these types of problems?

REPLY [30 votes]: This question is several years old. Nevertheless, I'm bothered that none of the answers provided were at an introductory level. Let me then contribute an answer, which is not rigorous, but will serve as a general heuristic for students who have only recently encountered fractional exponents with real numbers.

We can interpret a base raised to a simplified$^\dagger$ fractional exponent with this heuristic:
  $$x^{\frac ab} = x^{\frac{\text{'power'}}{\text{'root'}}}$$
  That's saying that $a$ acts like a standard integer power and $b$ acts like a standard integer root.$^{\dagger\dagger}$

Your example, $(-5)^{2/3}$, can be interpreted as squaring $-5$ and then taking the third root. Or, in the opposite order, taking the cube root of $-5$ and then squaring that result.
\begin{align}
(-5)^{2/3} &= ((-5)^2)^{1/3} = \sqrt[3]{25} \approx 2.92 
\\\\
\text{or}&
\\\\
(-5)^{2/3} &= (\sqrt[3]{-5})^{2} \approx (-1.71)^2 \approx 2.92  
\end{align}
Notice that in this particular example our base was negative. Since the denominator of the fraction was odd, we were able to solve for a real number. If the denominator were even, though, we would have no real solution, since the even root of a negative number is undefined for real numbers. Instead, we would have to turn to complex numbers for a more adequate interpretation (see the accepted answer by Hurkyl).

$^\dagger$ The fractional exponent must be simplified for the upcoming process to make sense. To see why, consider the example $(-8)^\frac 24$. What happens if you don't simplify? If you do?
$^{\dagger\dagger}$ We're assuming $a$ and $b$ are integers such that $a/b$ is  a rational number. That's likely what a student has seen when first encountering fractional exponents. If $a,b$ are not integers, then the meaning is less obvious.