Complex Numbers And Quadratic Equations question 220
Question: $ {{(-\sqrt{3}+i)}^{53}} $ where $ i^{2}=-1 $ is equal to [AMU 2000]
Options:
A) $ 2^{53}(\sqrt{3}+2i) $
B) $ 2^{52}(\sqrt{3}-i) $
C) $ 2^{53}( \frac{\sqrt{3}}{2}+\frac{1}{2}i ) $
D) $ 2^{53}(\sqrt{3}-i) $
Show Answer
Answer:
Correct Answer: C
Solution:
$ {{(-\sqrt{3}+i)}^{53}} $ $ =2^{53}{{( \frac{-\sqrt{3}}{2}+\frac{i}{2} )}^{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}[ \frac{\sqrt{3}}{2}+i\frac{1}{2} ] $ .
BETA
