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Square Root of a Complex Number

If 1,ω ,ω 2...

Question

If $1, ω, ω^{2}$ are the cube roots of unity, then the value of $(2 + ω) (2 + ω^{2}) (2 + ω^{10}) (2 + ω^{11})$ is

A

7

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B

49

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C

64

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D

9

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Solution

The correct option is D 9
$(2 + w) (2 + w^{2}) (2 + w^{10}) (2 + w^{11})$
$= (2 + w) (2 + w^{2}) (2 + w^{9} . w) (2 + w^{9} . w^{2})$
$= (2 + w) (2 + w^{2}) (2 + w) (2 + w^{2})$
$= ((2 + w) (2 + w^{2}))^{2}$
$= [4 + 2 (w + w^{2}) + w^{3}]^{2}$
$= [4 + 2 (- 1) + 1]^{2}$
$= (3)^{2}$
$= 9$

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