A) \[{{2}^{53}}(\sqrt{3}+2i)\]
B) \[{{2}^{52}}(\sqrt{3}-i)\]
C) \[{{2}^{53}}\,\left( \frac{\sqrt{3}}{2}+\frac{1}{2}i \right)\]
D) \[{{2}^{53}}(\sqrt{3}-i)\]
Correct Answer: C
Solution :
\[{{(-\sqrt{3}+i)}^{53}}\]\[={{2}^{53}}{{\left( \frac{-\sqrt{3}}{2}+\frac{i}{2} \right)}^{53}}\] = \[{{2}^{53}}{{(\cos {{150}^{o}}+i\sin {{150}^{o}})}^{53}}\] \[={{2}^{53}}[\cos ({{150}^{o}}\times 53)+i\sin ({{150}^{o}}\times 53)]\] \[={{2}^{53}}[\cos (22\pi +{{30}^{o}})+i\sin (22\pi +{{30}^{o}})]\] \[={{2}^{53}}[\cos {{30}^{o}}+i\sin {{30}^{o}}]\]\[={{2}^{53}}\left[ \frac{\sqrt{3}}{2}+i\frac{1}{2} \right]\].
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