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Question

If α and β are the complex cube roots of unity, then α4+β4+α1+β1.

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Solution

α and β are the complex cube roots of unity.

Hence,
α=ω,β=ω2

Since,
α4+β4+α1+β1

α4+β4+1α+1β

α4+β4+α+βαβ

Therefore,
ω4+(ω2)4+ω+ω2ωω2

ω+ω8+1ω3

ω+ω2+11

11

2

Hence, this is the answer.

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