Answer to Question #349058 in Complex Analysis for Mapula Advice

Question

Which of the following is a solution of 2z3+16i=0 ?

Answer 1

$2z^3+16i=0$

$z^3=-8i$

The polar form of $- 8 i$  is $8(\cos(-\dfrac{\pi}{2})+i\sin(-\dfrac{\pi}{2})).$

$k=0:$

$\sqrt[3]{8}(\cos(\dfrac{-\pi/2+2\pi(0)}{3})+i\sin(\dfrac{-\pi/2+2\pi(0)}{3}))$

$=\sqrt{3}-i$

$k=1:$

$\sqrt[3]{8}(\cos(\dfrac{-\pi/2+2\pi(1)}{3})+i\sin(\dfrac{-\pi/2+2\pi(1)}{3}))$

$=i$

$k=2:$

$\sqrt[3]{8}(\cos(\dfrac{-\pi/2+2\pi(2)}{3})+i\sin(\dfrac{-\pi/2+2\pi(2)}{3}))$

$=-\sqrt{3}-i$

The solutions are $\{-\sqrt{3}-i, i, \sqrt{3}-i\}.$