The value of $\large[\frac{-1+i\sqrt{3}}{2}]^{100}+[\frac{-1-i\sqrt{3}}{2}]^{100}$ is

Question

\[\begin{array}{1 1}(1)2&(2)0\\(3)-1&(4)1\end{array}\]

Answer 1

If $\omega$ is a cube root of unity then

$\omega^3=1$ and $1+\omega +\omega^2=0$

$\omega=-\large\frac{1}{2} $$ + i \large\frac{\sqrt 3}{2}$ and $\omega^2=-\large\frac{1}{2} - i \large\frac{\sqrt 3}{2}$

$\large[\frac{-1+i\sqrt{3}}{2}]^{100}+[\frac{-1-i\sqrt{3}}{2}]^{100}$

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

$\qquad=\omega^{100}+\omega^{200}$

$\qquad=\omega^{99}+\omega^{198}. \omega^{2}$

$\qquad=(\omega^{3})^{33}.\omega+(\omega^{3})^{66}. \omega^{2}$

$\qquad= (\omega^3)^{33}. \omega +( \omega^3)^{66}. \omega^{2}$

$\qquad=\omega+\omega^2 (\omega^3=1)$

$\qquad=-1$

Hence 3 is the correct answer.