(A) Let $\omega = \frac{-1 + i\sqrt{3}}{2}$ be the complex cube root of unity.Then $\omega^2 = \frac{-1 - i\sqrt{3}}{2}$.The expression becomes $\omeg

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(A) Let $\omega = \frac{-1 + i\sqrt{3}}{2}$ be the complex cube root of unity.Then $\omega^2 = \frac{-1 - i\sqrt{3}}{2}$.The expression becomes $\omega^{18} + (\omega^2)^{18} = \omega^{18} + \omega^{36}$.Since $\omega^3 = 1$, we have $\omega^{18} = (\omega^3)^6 = 1^6 = 1$.Similarly, $\omega^{36} = (\omega^3)^{12} = 1^{12} = 1$.Thus, $1 + 1 = 2$.

Class 11 Mathematics 4-1.Complex numbers De Moivre's theorem and Roots of unity

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The value of $(\frac{-1 + i\sqrt{3}}{2})^{18} + (\frac{-1 - i\sqrt{3}}{2})^{18}$ is

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A
$2$
B
$35$
C
$\sqrt{3}$
D
$\frac{\sqrt{3}}{2}$

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