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N[(-8)^(1/3)] gives 1. + 1.73205 i.

Since $(-8)^{\frac{1}{3}}$ has three possible solutions, one real and other two are complex.

So, why do not mathematica provides all the three solutions, and if it has to definitely choose only one solution then why not the REAL one ? Just curious !

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As Bob Hanlon said, you can use CubeRoot to get one root.

According to you question (the tree possible solutions) you can also simply use this:

Solve[x^3 == 8] // N
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