Find Roots of Complex Numbers in Polar Form
-The complex square roots of $4 + 4 \sqrt { 3 }$ i (rectangular form)
A) $\sqrt { 6 } + \sqrt { 2 } i , - \sqrt { 6 } - \sqrt { 2 } i$
B) $\sqrt { 6 } - \sqrt { 2 } i , - \sqrt { 6 } + \sqrt { 2 } i$
C) $\frac { \sqrt { 2 } } { 2 } + \frac { \sqrt { 6 } } { 2 } \mathrm { i } , - \frac { \sqrt { 2 } } { 2 } - \frac { \sqrt { 6 } } { 2 } \mathrm { i }$
D) $- \sqrt { 6 } - \sqrt { 2 } i , \sqrt { 6 } - \sqrt { 2 } i$

Question

Quizplus · Accepted Answer

The complex number $4 + 4\sqrt{3}i$ can be converted to polar form to find its roots. Its magnitude is $\sqrt{4^2 + (4\sqrt{3})^2} = 8$, and its argument is $\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$. The square roots in polar form are found by taking the square root of the magnitude, $\sqrt{8} = 2\sqrt{2}$, and halving the argument, $\frac{\pi}{6}$. Converting these back to rectangular form gives $\sqrt{6} + \sqrt{2}i$ and $-\sqrt{6} - \sqrt{2}i$, which matches choice A.

Find Roots of Complex Numbers in Polar Form -The Complex 4+434 + 4 \sqrt { 3 }4+43​

Find Roots of Complex Numbers in Polar Form
-The Complex $4 + 4 \sqrt { 3 }$