Question

${(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)}^{1000}$ is equal to

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

Answer 1

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

We have,

${(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)}^{1000}$

$\Rightarrow {(\cos \frac{2\pi }{3}+\sin \frac{2\pi }{3}i)}^{1000}$

$\Rightarrow (\cos \frac{1000\times 2\pi }{3}+\sin \frac{1000\times 2\pi }{3}i)$

$\Rightarrow (\cos \frac{2\pi }{3}+\sin \frac{2\pi }{3}i)$

$\Rightarrow (-\frac{1}{2}+\frac{\sqrt{3}}{2}i)$

$\Rightarrow \frac{-1+\sqrt{3}i}{2}$

Hence, this is the answer.