Find all specified roots.
-Cube roots of $\mathrm { i }$.
A) $\frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \mathrm { i }$
B) $\frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i }$, i
C) $\frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } + \frac { 1 } { 2 } \mathrm { i } , - \mathrm { i }$
D) $\frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i }$, i

Question

Quizplus · Accepted Answer

The cube roots of $i$ can be found by converting $i$ into polar form and using De Moivre's Theorem. The principal cube root of $i$ is $\frac{\sqrt{3}}{2} + \frac{1}{2}i$, and the other two roots are found by adding $2\pi/3$ and $4\pi/3$ to the argument, resulting in the three roots listed in option C.

Find All Specified Roots i\mathrm { i }i A) 32−12i,−32−12i,−i\frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \mathrm { i }23​​−21​i,−23​​−21​i,−i

Find All Specified Roots $\mathrm { i }$
A) $\frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \frac { \sqrt { 3 } } { 2 } - \frac { 1 } { 2 } \mathrm { i } , - \mathrm { i }$