Wolfram Alpha wrong answers on $(-8)^{1/3}$ and more? [duplicate]

Wolfram Alpha doesn't give $-2$ for $(-8)^{1/3}$, and it absolutely fails to draw $f(x)=x^{1/3}$ - does anyone know why? Am I missing something very 'deep' Wolfram Alpha is trying to teach me?

Here's what the graph should look like:

And here's what Wolfram Alpha draws:

marked as duplicate by Mark McClure, Kamil Jarosz, user228113, user223391, Eric WofseyMar 11 '16 at 21:13

• Yes. You are missing the fact that cubic root multi-valued (3 values) and W|A gives you the principle value while the other program uses a simpler convention and spits out a non-complex value. Try plotting $x^{1/2}$ instead in both to understand the difference deeper. – A.S. Mar 11 '16 at 8:29
• You can get the plot you expect by entering y=Abs(x)^(1/3)*Signum(x) into W/A. – LouisB Mar 11 '16 at 8:50
• You can enter cbrt(x) for the cube root function in Wolfram|Alpha, which (as implemented in that particular piece of software) is different from $x^{1/3}$. – Mark McClure Mar 11 '16 at 12:56

What you are probably thinking about is that for solving the equation $$z^k = a$$
we often have many solutions (for $z$) in the complex plane. More to the point, for a function one wants to assign one out-value for each in-value. Choosing a particular way to assign function values is called picking a branch for a function. For some functions there exist widely used conventions how to pick these branches and they get called principal branch. Integer roots and logarithms are famous examples of functions which have a principal branch.