Complex equation

The problem says to solve the given complex equation:

$$z^4-\left[ \frac{\sqrt3}{2}i^{21}+ \frac{\sqrt3}{2}i^{9}+ \frac{8}{(1+i)^6}\right]^9=0$$

The solution is this:

$$\cos\left( \frac{\pi}{8}+\frac{k\pi}{2} \right)+i\sin\left( \frac{\pi}{8}+\frac{k\pi}{2} \right) ,k=0,1,2,3$$

My problem is that I can't get this solution, I've tried multiple times to solve it but I always end up getting this solution:
$$z^4=(i+\sqrt3*i)^9$$

Could you help me with this? Thanks in advance.