Question

The value of $\left(\frac{-1+\sqrt{-3}}{2}\right)^{100}+\left(\frac{-1-\sqrt{-3}}{2}\right)^{100}$ is

(A) 0

(B) 1

(C) -1

(D) 2

Answer 1

$\left(\frac{-1+\sqrt{-3}}{2}\right)^{100}+\left(\frac{-1-\sqrt{-3}}{2}\right)^{100}$

$=(\omega)^{100}+\left(\omega^2\right)^{100}$

$=\left(\omega^3\right)^{39} \omega+\left(\omega^3\right)^{66}\omega^2$

$=\omega+\omega^2$$ $$\quad\left(\because \omega^3=1\right)$

$=-1$

So, The correct option will be (C).