Complex Numbers And Quadratic Equations question 252
Question: $ {{( -\frac{1}{2}+\frac{\sqrt{3}}{2}i )}^{1000}}= $
Options:
A) $ \frac{1}{2}+\frac{\sqrt{3}}{2}i $
B) $ \frac{1}{2}-\frac{\sqrt{3}}{2}i $
C) $ -\frac{1}{2}+\frac{\sqrt{3}}{2}i $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Here $ -\frac{1}{2}+\frac{1}{2}i\sqrt{3} $ is one of the two imaginary cube root of unity. If we denote it by $ \omega $ . Then $ {{\omega }^{1000}}={{\omega }^{999}}\omega ={{({{\omega }^{3}})}^{333}}\omega =\omega =-\frac{1}{2}+\frac{\sqrt{3}}{2}i $ .
BETA
