Answer to Question #319449 in Complex Analysis for Zizu

Question

Calculate z^6=64

Answer 1

$z^6=64\Rightarrow z=\sqrt[6]{64}e^{\frac{i\cdot 2\pi n}{6}},n=0,1,2,3,4,5\\z_0=2\\z_1=2e^{\frac{\iota \pi}{3}}=2\left( \cos \frac{\pi}{3}+i\sin \frac{\pi}{3} \right) =1+i\sqrt{3}\\z_2=2e^{\frac{2\pi i}{3}}=2\left( \cos \frac{2\pi}{3}+i\sin \frac{2\pi}{3} \right) =-1+i\sqrt{3}\\z_3=2e^{\pi i}=2\left( \cos \pi +i\sin \pi \right) =-2\\z_4=2e^{\frac{4\pi \iota}{3}}=2\left( \cos \frac{4\pi}{3}+i\sin \frac{4\pi}{3} \right) =-1-i\sqrt{3}\\z_5=2e^{\frac{5\pi i}{3}}=2\left( \cos \frac{5\pi}{3}+i\sin \frac{5\pi}{3} \right) =1-i\sqrt{3}$