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Question
If ω is a complex cube root of unity, find the value of (1 − ω − ω2)3 + (1 − ω + ω2)3
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Solution
Since ω is the complex cube root of unity,
ω3 = 1 and 1 + ω + ω2 = 0
∴ 1 + ω = – ω2, 1 + ω2 = – ω and ω + ω2 = – 1.
(1 − ω − ω2)3 + (1 − ω + ω2)3
= [1 – (ω + ω2)]3 + [(1 + ω2) – ω]3
= [1 – (–1)]3 + (–ω – ω)3
= 23 + (– 2ω)3
= 8 – 8ω3
= 8 – 8(1)
= 0
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