AP EAMCET · Maths · Complex Number
If \(1, \omega, \omega^2\) denote the cube roots of unity, then the value of \(\left(1-\omega+\omega^2\right)^5+\left(1+\omega-\omega^2\right)^5\) is
- A \(32 \omega^2\)
- B \(32 \omega\)
- C \(-32\)
- D \(32\)
Answer & Solution
Correct Answer
(D) \(32\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & 1+\omega+\omega^2=0 \\ & \left(1-\omega+\omega^2\right)^5+\left(1+\omega-\omega^2\right)^5 \\ & =(-2 \omega)^5+\left(-2 \omega^2\right)^5 \\ & =-32\left(\omega^5+\omega^{10}\right)=-32\left(\omega^2+\omega\right)=32\end{aligned}\)
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