[tex]z=2\mathrm{cis}\dfrac\pi6=2\left(\cos\dfrac\pi6+i\sin\dfrac\pi6\right)[/tex]
By DeMoivre's theorem, you have
[tex]z^5=2^5\mathrm{cis}\dfrac{5\pi}6=32\left(\cos\dfrac{5\pi}6+i\sin\dfrac{5\pi}6\right)[/tex]
[tex]\implies z^5=32\left(-\dfrac{\sqrt3}2+\dfrac12i\right)=-16\sqrt3+16i[/tex]
