To raise a complex number to a power, we use the De Moivre's Theorem.
Given,
[tex]z=2cis\frac{\pi}{6}[/tex]We can expland and write in >>>
[tex]\begin{gathered} 2(\cos \theta+i\sin \theta) \\ =2(\cos \frac{\pi}{6}+i\sin \frac{\pi}{6}) \end{gathered}[/tex]De Moivre's theorem tells us >>>>
[tex]\lbrack r(\cos \theta+i\sin \theta)\rbrack^n=r^n(\cos n\theta+i\sin n\theta)[/tex]Let's raise the complex number given to the fifth power >>>>
[tex]\begin{gathered} \lbrack2(\cos \frac{\pi}{6}+i\sin \frac{\pi}{6})\rbrack^5 \\ =2^5(\cos \frac{5\pi}{6}+i\sin \frac{5\pi}{6}) \\ =32(-\frac{\sqrt3}{2}+i\frac{1}{2}) \\ =-16\sqrt[]{3}+16i \end{gathered}[/tex]