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Question: Answered & Verified by Expert
$\left(\frac{-1+i \sqrt{3}}{2}\right)^{20}+\left(\frac{-1-i \sqrt{3}}{2}\right)^{20}=$
MathematicsComplex NumberJEE Main
Options:
  • A $20 \sqrt{3} i$
  • B $1$
  • C $\frac{1}{2^{19}}$
  • D $-1$
Solution:
1885 Upvotes Verified Answer
The correct answer is: $-1$
$\begin{aligned} & \text { As } \frac{-1+i \sqrt{3}}{2}=\omega \text { and } \frac{-1-i \sqrt{3}}{2}=\omega^2 \\ & \therefore(\omega)^{20}+\left(\omega^2\right)^{20}=\omega^{18} \cdot \omega^2+\omega^{39} \cdot \omega=\omega^2+\omega=-1\end{aligned}$

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