(C) Let $z = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$. This is the complex cube root of unity,denoted by $\omega$. We know that $\omega^3 = 1$. We need to

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(C) Let $z = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$. This is the complex cube root of unity,denoted by $\omega$. We know that $\omega^3 = 1$. We need to calculate $\omega^{1000}$. Since $1000 = 3 \times 333 + 1$,we have $\omega^{1000} = (\omega^3)^{333} \times \omega^1$. Substituting $\omega^3 = 1$,we get $\omega^{1000} = (1)^{333} \times \omega = \omega$. Therefore,$\omega^{1000} = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$.

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

Easy

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

A
$\frac{1}{2} + \frac{\sqrt{3}}{2}i$
B
$\frac{1}{2} - \frac{\sqrt{3}}{2}i$
C
$-\frac{1}{2} + \frac{\sqrt{3}}{2}i$
D
$\text{None of these}$

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4-1.Complex numbers

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De Moivre's theorem and Roots of unity

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

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