( - 1 2 + 3 2 i )^ 1000 = - Vidyadip

MCQ

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

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

✓

Answer

Correct option: C.

$ - \frac{1}{2} + \frac{{\sqrt 3 }}{2}i$

c
(c) Here $ - \frac{1}{2} + \frac{1}{2}i\sqrt 3 $ is one of the two imaginary cube root of unity. If we denote it by $\omega $.
Then ${\omega ^{1000}} = {\omega ^{999}}\omega = {({\omega ^3})^{333}}\omega = \omega = - \frac{1}{2} + \frac{{\sqrt 3 }}{2}i$.

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