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I am confused about the following problem. With $w=se^{i{\phi}}$, where $s\ge 0$ and $\phi \in \mathbb{R}$, solve the equation $z^n=w$ in $\mathbb{C}$ where $n$ is a natural number. How many solutions are there?

Now my approach is simply taking the $n$-th root which gives

$$z=\sqrt[n]{s}e^{\frac{i\varphi}{n}}$$

However, it seems that this problem is asking us to show the existance of the $n$-th root. Can I assume that the $n$-th root of a complex number already exists? Moreoover, would I be correct to say that there is only one solution which is given above?