Find all specified roots.
-Eighth roots of $i$.
A) $1 , - 1 , i , - i$
B) $\frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { 2 } } \mathrm { i } , \frac { 1 } { \sqrt { 2 } } - \frac { 1 } { \sqrt { 2 } } \mathrm { i } , - \frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { 2 } } \mathrm { i } , - \frac { 1 } { \sqrt { 2 } } - \frac { 1 } { \sqrt { 2 } } \mathrm { i }$
C) $\operatorname { cis } \frac { \pi } { 16 } , \operatorname { cis } \frac { 5 \pi } { 16 } , \operatorname { cis } \frac { 9 \pi } { 16 } , \operatorname { cis } \frac { 13 \pi } { 16 } , \operatorname { cis } \frac { 17 \pi } { 16 }$, cis $\frac { 21 \pi } { 16 } , \operatorname { cis } \frac { 25 \pi } { 16 } , \operatorname { cis } \frac { 29 \pi } { 16 }$
D) $1 , - 1 , i , - i , \frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { 2 } } \mathrm { i } , \frac { 1 } { \sqrt { 2 } } - \frac { 1 } { \sqrt { 2 } } \mathrm { i } , - \frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { 2 } } \mathrm { i } , - \frac { 1 } { \sqrt { 2 } } - \frac { 1 } { \sqrt { 2 } } \mathrm { i }$

Question

Quizplus · Accepted Answer

The eighth roots of $i$ can be found using the formula for the $n$th roots of a complex number, which in polar form is $r(\cos(\theta + 2k\pi) + i\sin(\theta + 2k\pi))^{1/n}$, where $k = 0, 1, ..., n-1$. For $i$, $r = 1$ and $\theta = \pi/2$, so the eighth roots are given by $\operatorname {cis} \frac{\pi/2 + 2k\pi}{8}$ for $k = 0$ to $7$, which simplifies to the options in choice C.

Find All Specified Roots iii A) 1,−1,i,−i1 , - 1 , i , - i1,−1,i,−i

Find All Specified Roots $i$
A) $1 , - 1 , i , - i$