Solve: $z^4 +2\sqrt3 +2i = 0$

Question

I'm already trying to solve this exercise for $20$ minutes, no luck. I got up to here:

$z^4 = -2i -2\sqrt3 = -2(\sqrt3 + i)$ but it's impossible to compute from here.

Also I tried: $z^8 = -2\sqrt3 -2i \rightarrow$ $z^8 = (4*3) -4 = 8 \rightarrow $ $z^8 = 8$

$z^8 = {{1}\over{8}}\text{Cis}(\pi)$ and from there to solve.

Write $-2i-2\sqrt 3$ in its polar form and use the first part of this answer. — Git Gud, Jul 02 '14 at 12:56

score 3 · Accepted Answer · answered Jul 02 '14 at 12:56

3

Hint: $$z^{4}=4\left(-\frac{\sqrt{3}}{2}-\frac{1}{2}i\right)=4e^{-\frac{5}{6}\pi i}$$

answered Jul 02 '14 at 12:56

Dario

Just solved it, thanks. – Ilan Aizelman WS Jul 02 '14 at 13:00
What's r? I know that it is $r^4 = 4$, so I keep it $r = ^{4}\sqrt4$? – Ilan Aizelman WS Jul 02 '14 at 13:09
And I keep that like that? – Ilan Aizelman WS Jul 02 '14 at 13:12

1 Answers1