If ω is the complex cube root of unity, find the value of (1 – ω

1.	If ω is the complex cube root of unity, find the value of (1 – ω – ω2)3 + (1 – ω + ω2)3
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

If ω is the complex cube root of unity, find the value of (1 – ω – ω2)3 + (1 – ω + ω2)3

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