Question

Prove that (1-ω + ω 2 ) 5 + (1 + ω - ω 2 ) 5 = 32

Answer 1

(1-ω+ω²)⁵+(1+ω-ω²)⁵
={(1+ω²)-ω}⁵+{(1+ω)-ω²}⁵
=(-ω-ω)⁵+(-ω²-ω²)⁵ [∵, 1+ω+ω²=0; ∴, 1+ω=-ω²; ∴, 1+ω²=-ω]
=(-2ω)⁵+(-2ω²)⁵
=-32ω⁵+(-32ω¹⁰)
=-32ω³.ω²-32ω³.ω³.ω³.ω
=-32ω²-32ω
=-32(ω+ω²)
=-32(-1) [∵, 1+ω+ω²=0; ∴, ω+ω²=-1]
=32 (Proved)