Complex root won't work So I'm trying to get this:
http://www.wolframalpha.com/input/?i=%288*sqrt%283%29%29%2F%28z%5E4%2B8%29%3Di
And I've calculated $z^4=16 \left( \cos (\frac{- \pi}{3})+ \sin ( \frac{- \pi}{3}) \right)$
So I'm trying to find the roots using the formula:
$ r^{1/n} = \left( \cos ( \frac { \theta + 2 \pi \cdot k}{n}) + i \cdot \sin ( \frac{ \theta + 2 \pi \cdot k}{n}) \right) $
But my result does not equal. Take a look:
http://www.wolframalpha.com/input/?i=16%5E%281%2F4%29*%28cos%28%28%28-pi%2F12%29%29%2F4%29%2Bi*sin%28%28%28-pi%2F12%29%29%2F4%29%29
This is for $k=0$.
What am I doing wrong?

From what I can tell, your error is in determining the argument of $z^4$.
I would start off by writing the reciprocal of both sides giving:
$$\frac{z^4+8}{8\sqrt{3}}=-i$$
Solving above equation for $z$ gives:
$$z^4=-8-8\sqrt{3}i$$
Thus, modulus of $z^4$ is $16$, as you correctly wrote. Your error is in determining the argument, you have to consider where $z^4$ lies in the complex plane. Since both the real and imaginary components are negative, it is in 3rd quadrant.

So, to find the arg, we take:
$$\pi+arctan(\frac{-8\sqrt{3}}{-8})=\frac{4\pi}{3}$$
Thus $z^4=16cis(\frac{4\pi}{3})$. Work from that and you should get the correct answer.
