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Question
If ω is a complex cube root of unity, find the value of (1 + ω)(1 + ω2)(1 + ω4)(1 + ω8)
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Solution
ω is the 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, ∴ ω4 = ω]
= (– ω2)(– ω)(– ω2)(– ω)
= ω6
= (ω3)2
= (1)2
= 1
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