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:
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The principal value of the cube root is given by Mathematica or Wolfram Alpha. If you really just want the real value, input with Sign[x] Abs[x]^(1/3)
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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.
What you are seeing is that every non-zero number has three distinct "cuberoots". In the first picture you've posted, the software very nicely graphed the real cuberoots. The problem is that Wolfram|Alpha is drawing a different branch of the cuberoot than you are used to, hence the labels of real and imaginary parts on the graph.