How can I show the $n$ distinct $n$th roots of 1 are equally spaced around a circle centered at the origin with radius 1?

Question

My work:

To show that the radius = 1 centered at the origin, I can employ the distance formula: $\sqrt{(r^{1/n}cos((\theta + 2k\pi)/n)-0)^2+(r^{1/n}isin((\theta + 2k\pi)/n)-0)^2}=\sqrt{r^{2/n}[cos^2((\theta + 2k\pi)/n)-sin^2((\theta + 2k\pi)/n)]}.$

However, I do not know where to go from here.

Answer 1

Use the polar form, so one root is at $re^{i\theta}$. Raise that to the $n^{th}$ power, getting $r^ne^{in\theta}$. You need $r^n=1, n\theta=2k\pi$