The domain of cubic root

Question

The domain of cubic root and in general $(2n-1)$ th root is $\mathbb{R}$. But Wolframalpha says the domain of cubic root is all non-negative real numbers. Also Matlab return 0.5000 + 0.8660i for (-1)^(1/3) and return 0.3969 + 0.6874i for (-0.5)^(1/3) that have an imaginary part. Although Excel return -1 and -0.7937. What is the problem?

Answer 1

There is no problem. As Wolfram Alpha writes it returns the principal cube root (as does Matlab). And Wolfram Alpha hints that you can Use the real‐valued root instead.

There a three (complex) cubic roots for a number. If you look at the diagram for the input (-1)^(1/3) you see the principal root, its conjugate and your real root.

Answer 2

The $n$-th root functions are multivalued functions, there are $n$ different function branches to pick from.

One does this because on that restriction the function is singlevalued and easier to work with (we have a naming problem here: a function with many values is a relation and not really a function).

In your case $n = 3$, there are three different ones. And your mentioned programs pick different ones.