Question

If w is the complex cube root of unity, find the value of
i) (1 - w - w²)³ + (1 - w + w²)³
ii) (1 + w)(1 + w²)(1 + w⁴)(1 + w⁸)

Answer 1

Answer:

i)-16

ii) 1

Step-by-step explanation:

Concept:

If ω is a cube root of unity

then

1+ω+ω²=0

and

ω³ = 1

i)i) (1 +ω- ω²)³ + (1 - ω + ω²)³

=(- ω² - ω²)³ + ( - ω - ω)³

=(- 2ω²)³ + ( - 2ω )³

=(- 8(ω³)²) + ( - 8 ω³)

=(- 8(1)²) + ( - 8 (1))

= - 8 - 8

= -16

ii) (1 + ω)(1 + ω²)(1 + ω⁴)(1 + ω⁸)

=(1 + ω)(1 + ω²)(1 + ω³.ω)(1 + ω³.ω³.ω²)

=(1 + ω)(1 + ω²)(1 + ω)(1 + ω²)

=[(1 + ω)(1 + ω²)]²

=[1 + ω² + ω +ω³]²

=[1 + (ω² + ω) +1]²

=[1 + (-1) +1]²

=[1]²

=1