Problem 1.19

Fix a positive integer n and a complex number w. Find all solutions to \(z^n=w.\)
(Hint:write w in terms of polar coordinates.)

Let \(n \in \mathbb{Z}\) and \(w \in \mathbb{C}.\) So, in polar coordinates, \(w=re^{i\theta}\) for some \(r,\theta \in \mathbb{R}.\) Notice that the \(r\) represents the modulus of the complex number and \(\theta\) is the argument. So, we have \(z^n=re^{i\theta}.\)
Geometrically, we are multiplying the lengths and add their angles.
We need to find values of \(z\) where it is multiplied by itself \(n\) times and is equal to \(re^{i\theta}.\)
So, let \(z=se^{i\theta}.\)
Thus, we have \(s^n=r.\)
Hence, \(s=\sqrt[n]{r}.\)
To solve for \(\theta\), we have \(n\theta= \theta +2\pi k\), for some \(k\in\mathbb{Z}.\)
Then \(\theta=\frac{\theta + 2 \pi k}{n}.\)
Therefore we have \(z=\sqrt[n]{r}e^{i\frac{\theta +2 \pi k}{n}}.\) By the Fundamental Theorem of Algebra, there are n solutions to \(z^n =w.\)