Sum

If ω is a complex cube root of unity, then prove the following: (ω^{2} + ω - 1)^{3} = – 8

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#### Solution

ω is a complex cube root of unity

∴ ω^{3} = 1 and 1 + ω + ω^{2} = 0

Also, 1 + ω^{2} = - ω, 1 + ω = - ω^{2}

and ω + ω^{2} = – 1

L.H.S. = (ω^{2} + ω - 1)^{3}

= (– 1 – 1)^{3}

= (– 2)^{3}

= – 8 = R.H.S.

Concept: Concept of Complex Numbers - Cube Roots of Unity

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