Question

If a complex number $z=\frac{\sqrt{3}}{2}+i\frac{i}{2}$ then $z^4$ is

• A. $2\sqrt{2}+2i$
• B. $-\frac{1}{2}+i\frac{\sqrt{3}}{2}$
• C. $\frac{\sqrt{3}}{2}-i\frac{1}{2}$
• D. $\frac{\sqrt{3}}{8}-i\frac{1}{8}$

Answer 1

The correct option is B $-\frac{1}{2}+i\frac{\sqrt{3}}{2}$

Given complex number is
$z=\frac{\sqrt{3}}{2}+i\left(\frac{1}{2}\right)=cos\frac{\pi }{6}+isin\frac{\pi }{6}=e^{\frac{i\pi }{6}}$
So, $z^4={\left(e^{i\frac{\pi }{6}}\right)}^4=e^{\frac{2\pi }{3}i}$
$=cos\frac{2\pi }{3}+isin\frac{2\pi }{3}$
=$-\frac{1}{2}+i\frac{\sqrt{3}}{2}$