Which of the following is a fourth root of fr | vedclass

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(B) The given complex number is $z = \frac{1}{2} + i\frac{\sqrt{3}}{2}$.In polar form,this is $z = \cos\left(\frac{\pi}{3}\right) + i\sin\left(\frac{\

Vedclass · Accepted Answer

$cis\left(\frac{\pi}{12}\right)$

Vedclass · Answer

(B) The given complex number is $z = \frac{1}{2} + i\frac{\sqrt{3}}{2}$.In polar form,this is $z = \cos\left(\frac{\pi}{3}ight) + i\sin\left(\frac{\pi}{3}ight) = cis\left(\frac{\pi}{3}ight)$.By De Moivre's Theorem,the $n$-th roots of $cis(	heta)$ are given by $cis\left(\frac{	heta + 2k\pi}{n}ight)$ for $k = 0, 1, 2, 3$.For the fourth root $(n=4)$ and $k=0$,we get $cis\left(\frac{\pi/3}{4}ight) = cis\left(\frac{\pi}{12}ight)$.Thus,$cis\left(\frac{\pi}{12}ight)$ is a fourth root of the given number.

Which of the following is a fourth root of $\frac{1}{2} + \frac{i\sqrt{3}}{2}$?

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