Question

(a+b)^2+(aw+bw^2)^2+(aw^2+bw)^2=6ab

Answer 1

Answer:

(a + b)² + (aω + bω²)² + (aω² + bω)² = 6ab

Step-by-step explanation:

(a+b)^2+(aw+bw^2)^2+(aw^2+bw)^2=6ab

LHS =

(a + b)² + (aω + bω²)² + (aω² + bω)²

= a² + b² + 2ab  + a²ω² + b²ω⁴ + 2abω³ + a²ω⁴ + b²ω² + 2abω³

= a² + b² + 2ab  + a²ω² + b²ω³.ω + 2abω³ + a²ω³.ω + b²ω² + 2abω³

ω³ = 1

= a² + b² + 2ab  + a²ω² + b²ω + 2ab + a²ω + b²ω² + 2ab

= a² + a²ω² + a²ω + b² + b²ω + b²ω² + 2ab + 2ab + 2ab

= a²(1 + ω² + ω) + b²(1 + ω + ω²) + 6ab

1 + ω + ω² = 0

= 0 + 0 + 6ab

= 6ab

= RHS

QED

Proved

(a + b)² + (aω + bω²)² + (aω² + bω)² = 6ab