Find all cube roots of the complex number. Leave answers in trigonometric form.
-$$4 - 4 i \sqrt { 3 }$$
A) $\sqrt [ 3 ] { 8 } \operatorname { cis } 100 ^ { \circ } , \sqrt [ 3 ] { 8 } \operatorname { cis } 220 ^ { \circ } , \sqrt [ 3 ] { 8 } \operatorname { cis } 340 ^ { \circ }$
B) $\sqrt { 4 } \operatorname { cis } 60 ^ { \circ } , \sqrt { 4 } \operatorname { cis } 180 ^ { \circ } , \sqrt { 4 } \operatorname { cis } 300 ^ { \circ }$
C) $\sqrt { 8 } \operatorname { cis } 100 ^ { \circ } , \sqrt { 8 } \operatorname { cis } 220 ^ { \circ } , \sqrt { 8 } \operatorname { cis } 340 ^ { \circ }$
D) $\sqrt [ 3 ] { 4 } \operatorname { cis } 80 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 200 ^ { \circ } , \sqrt [ 3 ] { 4 } \operatorname { cis } 320 ^ { \circ }$

Question

Quizplus · Accepted Answer

The complex number $4 - 4i\sqrt{3}$ can be converted to polar form as $r\operatorname{cis}\theta$, where $r = \sqrt{4^2 + (4\sqrt{3})^2} = 8$ and $\theta = \tan^{-1}\left(\frac{-4\sqrt{3}}{4}\right) = -60^\circ$. When finding cube roots, $r$ becomes $\sqrt[3]{8} = 2$ and $\theta$ is divided by 3 and adjusted by $120^\circ$ for each root, resulting in angles of $100^\circ, 220^\circ, 340^\circ$.

Find All Cube Roots of the Complex Number 4−4i34 - 4 i \sqrt { 3 }4−4i3​

Find All Cube Roots of the Complex Number $4 - 4 i \sqrt { 3 }$