cubic root of negative numbers

-8 has three cube roots: $ -2 $, $1 + i \sqrt{ 3 } $ and $1 - i \sqrt{ 3 } $. So you can't answer the question "Is $ \sqrt[3]{-8} $ real" without specifying which of them you're talking about.

For some reason, WolframAlpha is only giving $1 + i \sqrt{ 3 } $ as an answer -- that looks like a bug in WolframAlpha to me.

Although it's been two years since this question was asked, some folks might be interested to know that this behavior has been modified in WolframAlpha. If you ask for the cube root of a negative number, it returns the real valued, negative cube root. Here, I just asked for "cbrt -8", for example:

Note "the principal root" button. That allows you to toggle back to the original behavior. Near the bottom, we still see information on all the complex roots.

We can plot functions involving the cube root and solve equations involving the cube root and it consistently acts real valued. If you just type in an equation, it will solve it, plot both sides and highlight the intersections. Here's "cbrt(x)=sin(2x)"

See this. In particular, the prinicipal cube root has nonzero imaginary part.