ROUGH DRAFT authorea.com/106016

# Problem 1.19

[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 $$z^n =w.$$