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I am doing an exercise in a book and although I think the answer is intuitive I am clueless when I try to think of a good argument.

The problem is as follows:

Let $f_1(z),...f_m(z)$ be all the continuous single-valued branches of the function $f(z)$ Using the same cuts, find all the continuous single-valued branches of the function $h(z)=f(z)^n$ where n is a non-zero integer.

The answer is $f_1(z)^n,...f_m(z)^n$ so it makes scence but I have no good argument that sais these are all the branches (I think I can safely say that $f_1(z)^n,...f_m(z)^n$ are all branches, but why every branch is some $f_i^n ?)$

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