branch cut and branch points for $f(z)=[(z-1)(z-2)]^{1/3}$

Question

I am trying to identify a possible branch cut and branch points for $f(z)=[(z-1)(z-2)]^{1/3}$.

I am confused why this would even need a branch cut. My problem says it is multivalued, but it seems like it isn't. For example, a square root has a positive and negative answer, but the third root doesn't have that problem. Could anyone help me out with my misunderstanding?

If there are branch points they would probably be 1 and 2.

Thanks.