Complex Numbers And Quadratic Equations question 839
Question: $ {{(-1+i\sqrt{3})}^{20}} $ is equal to [RPET 2003]
Options:
A) $ 2^{20}{{(-1+i\sqrt{3})}^{20}} $
B) $ 2^{20}{{(1-i\sqrt{3})}^{20}} $
C) $ 2^{20}{{(-1-i\sqrt{3})}^{20}} $
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
Let $ z=-1+i\sqrt{3} $ , $ r=\sqrt{1+3}=2 $ $ \theta ={{\tan }^{-1}}( \frac{\sqrt{3}}{-1} )=\frac{2\pi }{3} $
$ \therefore z=2( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} ) $
$ \therefore {{(z)}^{20}}={{[ 2( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} ) ]}^{20}} $ $ =2^{20}{{( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} )}^{20}} $ $ =2^{20}{{( -\frac{1}{2}+i\frac{\sqrt{3}}{2} )}^{20}} $ .
BETA
