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Properties of Cube Root of a Complex Number

If ω is a c...

Question

If $ω$ is a cube root of unity, then $(1 + ω - ω^{2})^{5} + (1 - ω + ω^{2})^{5} =$

A

32

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B

31

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C

30

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D

28

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Solution

The correct option is A 32
$1 + w = - w^{2}$
$1 + w^{2} = - w$
Hence,
$(- 2 w^{2})^{5} + (- 2 w)^{5}$
$= - 2^{5} (w + w^{2})$
$= 32$

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