If ω is the complex cube root of unity, find the value of (1 – ω – ω^2)^3 + (1 – ω + ω^2)^3
If ω is the complex cube root of unity, find the value of (1 – ω – ω2)3 + (1 – ω + ω2)3
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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 – ω2)3 + (1 – ω + ω2)3
= [1 – (ω + ω2)]3 + [(1 + ω ) – ω2]3
= [1 – (-1)2] + (-ω – ω)3
= 23 + (-2ω)3
= 8 – 8ω3
= 8 – 8(1)
= 0
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