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nth Root of a Complex Number

If 1, ω, ω2, ...

Question

$I f 1, ω, ω^{2}, ω^{3} . . . . ., ω^{n - 1} a r e t h e n, n^{t h} r o o t s o f u n i t y, t h e n (1 - ω) (1 - ω^{2}) . . . . . (1 - ω^{n - 1}) e q u a l s$

A

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B

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C

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D

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Solution

The correct option is C
$S i n c e 1, ω, ω^{2}, ω^{3}, . . . . . ω^{(} n - 1) a r e t h e n, n^{t h} r o o t s o f u n i t y, t h e r e f o r e, w e h a v e t h e i d e n t i t y = (x - 1) (x - ω) (x - ω^{2}) . . . . . (x - ω^{n - 1}) = x^{n} - 1 o r (x - ω) (x - ω^{2}) . . . . . . (x - ω^{n - 1}) = \frac{x^{n} - 1}{x - 1} = x^{n - 1} + x^{n - 2} + . . . . + x + 1 P u t t i n g x = 1 o n b o t h s i d e s, w e g e t (1 - ω) (1 - ω^{2}) . . . . . (1 - ω^{n - 1}) = n$

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1, ω, ω^{2}, ω^{3} . . . . ., ω^{n - 1}

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