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Question
If ω is a complex cube root of unity, show that (1 + ω)3 − (1 + ω2)3 = 0
Solution
ω is a complex cube root of unity.
∴ ω3 = 1 and 1 + ω + ω2 = 0
∴ ω + ω2 = – 1, 1 + ω = – ω2 and 1 + ω2 = – ω.
(1 + ω)3 – (1 + ω2)3
= (– ω2)3 – (– ω)3
= – ω6 – (– ω3)
= – (ω3)2 + ω3
= – (1)2 + 1
= – 1 + 1
= 0
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