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Question
If ω is a complex cube root of unity, show that (2 − ω)(2 − ω2) = 7
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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
(2 – ω)(2 – ω2) = 7
= 4 – 2ω2 – 2ω + ω3
= 4 – 2(ω2 + ω) + ω3
= 4 – 2(– 1) + 1
= 4 + 2 + 1
= 7
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