TITLE: cubic root of negative numbers
QUESTION [31 upvotes]: Excuse my lack of knowledge and expertise in math,but to me it would came naturally that the cubic root of $-8$ would be $-2$ since $(-2)^3 = -8$.
But when I checked Wolfram Alpha for $\sqrt[3]{-8}$, real it tells me it doesn't exist.
I came to trust Wolfram Alpha so I thought I'd ask you guys, to explain the sense of that to me.

REPLY [3 votes]: In the years since the question was asked and answered, Wolfram introduced the $\operatorname{Surd}(x,n)$ function (Mathematica 9 circa $2012$, then Alpha) to designate the real single-valued $n^{th}$ root of $x$.
For example $\sqrt[3]{-8}$ and $\sqrt[5]{-243}$ result directly in $-2$ and $-3$ respectively:


The $\operatorname{Cbrt}$ function discussed in Mark McClure's answer - which had changed behavior around the same time to return the real cube root by default - appears to be identical to $\operatorname{Surd}(\,\cdot\,,3)$: