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.