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

Question

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

Aicharya · Accepted Answer

To Find 4th root of space $\frac{1}{2}+\frac{i \sqrt{3}}{2}$ $\begin{aligned} & 	ext { Let } Z=\frac{1}{2}+i \frac{\sqrt{3}}{2}=\cos \frac{\pi}{3}+i \sin \frac{\pi}{3} \ & \left.\quad=e^{i^{\frac{\pi}{3}}} \quad \quad \quad \because e^{i 	heta}=\cos 	heta+i \sin 	hetaight\} \ & 	herefore z^{\frac{1}{4}}=\left(e^{i \frac{\pi}{3}}ight)^{1 / 4}=e^{i \pi / 12} 	ext { or } \operatorname{cis} \frac{\pi}{12} \ & 	herefore 	ext { 4th root of } z 	ext { is cis } \frac{\pi}{12}\end{aligned}$

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

Solution