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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
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 - 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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