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  4. Least positive argument of the 4 text th root of the complex number 2-i √12 is

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Q. Least positive argument of the $4^{\text {th }}$ root of the complex number $2-i \sqrt{12}$ is

Complex Numbers and Quadratic Equations

Solution:

$z^{4}=2(1-\sqrt{3} i)=4\left(\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)$
$=4\left[\cos \left(-\frac{\pi}{3}\right)+i \sin \left(-\frac{\pi}{3}\right)\right]$
$z=\sqrt{2}\left[\cos \frac{2 m \pi-(\pi / 3)}{4}+i \sin \frac{2 m \pi-(\pi / 3)}{4}\right]$
For $m=1, z=\sqrt{2}\left[\cos \left(\frac{5 \pi}{12}\right)+i \sin \left(\frac{5 \pi}{12}\right)\right]$.