Find the indicated roots. Write the answer in a + bi form.
-Cube roots of $- 8 \mathrm { i }$
A) $- 2 i , \sqrt { 3 } + i , \sqrt { 3 } + i$
B) $2 i , - \sqrt { 3 } - i , \sqrt { 3 } - i$
C) $2 i , - \sqrt { 3 } + i , - \sqrt { 3 } + i$
D) $- 2 i , \sqrt { 3 } - i , \sqrt { 3 } - i$

Question

Quizplus · Accepted Answer

The cube roots of $-8i$ can be found by converting $-8i$ into polar form and then applying De Moivre's Theorem. The polar form of $-8i$ is $8(\cos(270^\circ) + i\sin(270^\circ))$, and its cube roots will have a magnitude of $8^{1/3} = 2$ and angles of $(270^\circ + 360^\circ k)/3$, where $k = 0, 1, 2$. This results in angles of $90^\circ$, $210^\circ$, and $330^\circ$, corresponding to the complex numbers $2i$, $-\sqrt{3} - i$, and $\sqrt{3} - i$, respectively.

Find the Indicated Roots −8i- 8 \mathrm { i }−8i A) −2i,3+i,3+i- 2 i , \sqrt { 3 } + i , \sqrt { 3 } + i−2i,3​+i,3​+i

Find the Indicated Roots $- 8 \mathrm { i }$ A) $- 2 i , \sqrt { 3 } + i , \sqrt { 3 } + i$