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in Complex Numbers by (32.0k points)
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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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Best answer

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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