( - 1 + i 3 )^ 20 is equal to - Vidyadip

MCQ

${( - 1 + i\sqrt 3 )^{20}}$ is equal to

A
${2^{20}}{( - 1 + i\sqrt 3 )^{20}}$
B
${2^{20}}{(1 - i\sqrt 3 )^{20}}$
C
${2^{20}}{( - 1 - i\sqrt 3 )^{20}}$
✓
None of these

✓

Answer

Correct option: D.

None of these

d
(d) Let $z = - 1 + i\sqrt 3 $, $r = \sqrt {1 + 3} = 2$
$\theta = {\tan ^{ - 1}}\left( {\frac{{\sqrt 3 }}{{ - 1}}} \right) = \frac{{2\pi }}{3}$
$\therefore \,z = 2\,\left( {\cos \frac{{2\pi }}{3} + i\sin \frac{{2\pi }}{3}} \right)$
$\therefore \,\,{(z)^{20}} = {\left[ {2\left( {\cos \frac{{2\pi }}{3} + i\sin \frac{{2\pi }}{3}} \right)} \right]^{20}}$
$ = {2^{20}}{\left( {\cos \frac{{2\pi }}{3} + i\sin \frac{{2\pi }}{3}} \right)^{20}}$$ = {2^{20}}{\left( { - \frac{1}{2} + i\frac{{\sqrt 3 }}{2}} \right)^{20}}$.

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