Question

The value of ${\left(\frac{-1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right)}^{1000}$is

• A. ${\omega }^3$
• B. ${\omega }^2$
• C. ${\omega }^3-\omega$
• D. $\omega$

Answer 1

The correct option is D

$\omega$

The explanation for the correct answer.

Find the value

Given: ${\left(\frac{-1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right)}^{1000}$

we know that $\left(\frac{-1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right)=\omega$ and ${\omega }^{3n}=1$

$\begin{array}{rcl}& \Rightarrow & {\mathrm{\omega }}^{1000}={({\mathrm{\omega }}^3)}^{{333}^{}}·\mathrm{\omega }\\ & \Rightarrow & {\mathrm{\omega }}^{1000}=1.\mathrm{\omega }\\ & \Rightarrow & {\mathrm{\omega }}^{1000}=\mathrm{\omega }\\ & \Rightarrow & \left(\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right)=\mathrm{\omega }\end{array}$

Hence $o\mathrm{ption}\left(\mathrm{D}\right)$ is the correct answer.