Question

If w is the cube root of unity. Prove that (1 - w + w²)⁶ + (1 + w - w²)⁶ = 128

Answer 1

Answer:

128

Step-by-step explanation:

Concept:

If ω is a cube root of unity

then

1+ω+ω²=0

and

ω³ = 1

( 1 - ω+ ω² )⁶ + ( 1 + ω - ω² )⁶

=( - ω - ω )⁶ + ( - ω² - ω² )⁶

=( - 2ω )⁶ + ( - 2ω² )⁶

$= 2^6 {\omega}^6+2^6 ({\omega^2})^6\\\\=64 {\omega^6}+64{\omega^{12}}\\\\=64 {(\omega^3)^2}+64{(\omega^3)4}\\\\\pi =64 {(1)^2}+64{(1)^4}$

=64 +64

=128