( - 3 + i)^ 53 where i^2 = - 1 is equal to - Vidyadip

MCQ

${( - \sqrt 3 + i)^{53}}$ where ${i^2} = - 1$ is equal to

A
${2^{53}}(\sqrt 3 + 2i)$
B
${2^{52}}(\sqrt 3 - i)$
✓
${2^{53}}\,\left( {\frac{{\sqrt 3 }}{2} + \frac{1}{2}i} \right)$
D
${2^{53}}(\sqrt 3 - i)$

✓

Answer

Correct option: C.

${2^{53}}\,\left( {\frac{{\sqrt 3 }}{2} + \frac{1}{2}i} \right)$

c
(c) ${( - \sqrt 3 + i)^{53}}$$ = {2^{53}}{\left( {\frac{{ - \sqrt 3 }}{2} + \frac{i}{2}} \right)^{53}}$
= ${2^{53}}{(\cos {150^o} + i\sin {150^o})^{53}}$
$ = {2^{53}}[\cos ({150^o} \times 53) + i\sin ({150^o} \times 53)]$
$ = {2^{53}}[\cos (22\pi + {30^o}) + i\sin (22\pi + {30^o})]$
$ = {2^{53}}[\cos {30^o} + i\sin {30^o}]$$ = {2^{53}}\left[ {\frac{{\sqrt 3 }}{2} + i\frac{1}{2}} \right]$.

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