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Question

Mathematics Question on Complex Numbers and Quadratic Equations

If ω\omega and ω2\omega^2 are complex cube roots of unity, then (1−ω+ω2)5+(1−ω2+ω)5(1-\omega+\omega^2)^5+(1-\omega^2+\omega)^5 is equal to

A

88

B

1616

C

3232

D

6464

Answer

3232

Explanation

Solution

(1−ω+ω2)5+(1−ω2+ω)5\left(1-\omega+\omega^{2}\right)^{5}+\left(1-\omega^{2}+\omega\right)^{5} =(−2ω)5+(−2ω2)5=\left(-2\omega\right)^{5}+\left(-2\omega^{2}\right)^{5} =−32(ω2+ω)=-32\left(\omega^{2}+\omega\right) =−32(−1)=32=-32\left(-1\right)=32.