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Cube Root of a Complex Number

If 'ω ' is ...

Question

If $^{'} ω^{'}$ is a complex cube root of unity, then $(1 - ω + w^{2})^{7} + (1 + ω - ω^{2})^{7} =$

A

2^{7}

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B

2^{6}

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C

2^{5}

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D

2^{4}

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Solution

The correct option is A $2^{7}$
$(1 - w + w^{2})^{7} + (1 + w - w^{2})^{7}$
$= [(1 + w^{2}) - w]^{7} + [(1 + w) - w^{2}]^{7}$
$= [- w - w]^{7} + [- w^{2} - w^{2}]^{7}$
$= (- 2 w)^{7} + (- 2 w^{2})^{7}$
$= 2^{7} (- w^{7} - w^{14})$
$= 2^{7} (- w^{6} . w - w^{12} . w^{2})$
$= 2^{7} (- w - w^{2})$
$= 2^{7} (1)$
$= 2^{7}$

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