Find all the solutions of $z^7=i$

Question

Find all the solutions of $z^7=i$.

The modulus is 1 and the argument is $\frac{\pi}{2}$, so

$z^7=i=1\cdot e^{(\frac{\pi}{2}+2\pi n)i}$

$z=i^{1/7}=e^{(\frac{\pi}{14}+\frac{2\pi n}{7})i}$ with $0\leq n <7$

So $\cos(\frac{\pi}{14}+\frac{2\pi n}{7})+i\, \sin(\frac{\pi}{14}+\frac{2\pi n}{7})$ by De Moivre?

I this right so far? In the answers it says the first root is $-i$ but I'm not getting this, any help?

Answer 1

You are doing fine: there should be seven distinct solutions, and you cannot really call any of them the first.

If $n=5$ then $\frac{\pi}{14}+\frac{2\pi n}{7} = \frac{3 \pi}{2}$

Or if $n=-2$ then $\frac{\pi}{14}+\frac{2\pi n}{7} = -\frac{\pi}{2}$

Answer 2

You are on the right tracks.

$-i$ can also be expressed as $e^{-\frac{\pi}{2}i}$ or $e^{\frac{3\pi}{2}}i$. Which is the case when $n = 5$.

The rest you can get by sticking in the values for $n$