If $z = r[\cosθ +i\sinθ]$, show that $w = \sqrt[n]{r}[\cos(θ/n) +i\sin(θ/n)]$

Question

a. If $z=r\left[\cos\theta +i\sin\theta \right]$, then $ w =\sqrt[n]{r}\left[\cos\left(\frac{\theta}{n}\right) +i\sin\left(\frac{\theta }{n}\right)\right] $ is an $n$-th root of $z$, where $r\geq 0$

Now my first week of abstract algebra just ended so I really haven’t learn too much yet.

I can see this looks like polar coordinates and that it might involve De Moivre's Theorem but I’ve never seen a problem like this before. If someone could start me off or tell me for sure what theorem to use I would appreciate it

b. Show that every $n$-th root of $z$ has the form $ζ kw$, where $ζ$ is a primitive $n$-th root of unity and $k = 0,1,2,\dotsc,n−1$.

I'm more concerned to learn how to do part a. than b.

Answer 1

Hint: Use Euler's Identity

$ e^{\mathbb{i} \theta} = \cos{\theta }+ \mathbb{i} \sin{\theta}$

Now,

$z=r\left[\cos\theta +i\sin\theta \right]$ $\rightarrow$ $z=re^{\mathbb{i} \theta}$

So,

$z^{\frac{1}{n}} = r^{\frac{1}{n}}e^{\frac{\mathbb{i} \theta}{n} } $

Then

$z^{\frac{1}{n}} = r^{\frac{1}{n}}\left[\cos\frac{\theta}{n} +i\sin\frac{\theta}{n} \right]$