Question

Calculate, $\sqrt[4]{4-1\frac{1}{2}-i\frac{\sqrt{3}}{2}}.$

Answer 1

$z=(\frac{-3}{2}-\frac{-i\sqrt{3}}{2}{)}^{1/4}$

$z^4=(\frac{-3}{2}-\frac{-i\sqrt{3}}{2})=r{e}^{i\theta }$

$r=\sqrt{\frac{9}{4}+\frac{3}{4}}=\sqrt{3}$

$\theta =\tan -1(\frac{\sqrt{3}}{2}\times \frac{2}{3})=\pi /6=7\pi /6$

$z^4=\sqrt{3}{e}^{i7\pi /6}$

$⟹z=\sqrt{3}{e}^{i7\pi /24}$

$⟹\sqrt{3}(\cos \frac{7\pi }{24}+i\sin \frac{7\pi }{24})$