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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)`
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) ...[∵ ω3 = 1, therefore ω4 = ω]
= (- ω2)(- ω)(- ω2)(- ω) = ω6 = (ω3)2 = (1)2 =1.
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