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Question
If `omega` is a complex cube root of unity, find the value of `(1 + omega)(1 + omega^2)(1 + omega^4)(1 + omega^8)`
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Solution
ω is a complex cube root of unity
∴ ω3 = 1 and 1 + ω + ω2 = 0
Also, 1 + ω2 = -ω, 1 + ω = -ω2 and ω + ω2 = –1
(1 + ω)(1 + ω2)(1 + ω4)(1 + ω8)
= (1 + ω)(1 + ω2)(1 + ω)(1 + ω2) ...[โต ω3 = 1, ∴ ω4 = ω]
= (- ω2) (- ω) (- ω2) (- ω)
= ω6
= (ω3)2
= (1)2
= 1
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