@anomaly, I don't think that you understand this question.
In any case, you did not answer my question.
$\left[2\exp\left(\dfrac{\pi i}{12}\right)\right]^4=16\exp\left(\dfrac{\pi i}{3}\right)=16\left(\dfrac{1}{2}+i~\dfrac{\sqrt3} {2}\right)$
I think that you should post the original question in exact wording.
You are taking the third root, but you should be taking the fourth root of $z^4$. The first root is $z$. All four roots can be found like this:
$$\left(16e^{\tfrac{i\pi}{3}+2ni\pi}\right)^{1/4}, n=0,1,2,3$$
For $n=0$, you have $z=2e^{\tfrac{i\pi}{12}}$.
For $n=1$, you have $2e^\tfrac{7i\pi}{12}$.
For $n=2$, you have ...
For $n=3$, you have ...
Complete the last two.