Find all the complex roots. Leave your answers in polar form with the argument in degrees.
-The complex cube roots of $- 8 \mathrm { i }$
A) $2 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 2 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) , 2 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$
B) $512 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 512 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) , 512 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$
C) $8 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 8 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) , 8 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$
D) $2 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right) , 2 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right) , 2 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$

Question

Quizplus · Accepted Answer

The complex number $-8i$ can be written in polar form as $8(\cos270^\circ + i\sin270^\circ)$. To find its cube roots, we divide the argument by 3 and take the cube root of the magnitude. The magnitude of the cube roots is $\sqrt[3]{8} = 2$, and the arguments are $90^\circ$, $90^\circ + 120^\circ = 210^\circ$, and $90^\circ + 240^\circ = 330^\circ$, resulting in the roots listed in option A.

Find All the Complex Roots −8i- 8 \mathrm { i }−8i

Find All the Complex Roots $- 8 \mathrm { i }$