Choosing multiple branch cuts

Question

We have $z^{\frac{1}{3}}$. I need to find three branch cuts of this function. I know branch cuts are made such that the function becomes single valued. However, I'm really uncertain on how to find the branch cuts for this function. I've read up on the topic and discovered that they can be chosen at random, depending on the problem you're dealing with. This doesn't seem helpful.
I have determined that the branch point for this function is rather obvious: it's $0$. How do I use this fact to produce three branch cuts? If I needed only one, then I would be able to determine one. How does it work for three? Thanks in advance.

Answer 1

I think you are confusing branches and branch points. The only branch points are $0,\infty$. To find three branches of the function, cut along $\mathbb R^-$. Then you have a standard branch $z\mapsto |z|e^{i \theta/3}$ where $\theta=\arg z\in(-\pi,\pi)$. The other two are obtained by multiplying by a cube root of unity.