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nth Root of a Complex Number

If ω and ...

Question

If $ω$ and $ω^{2}$ are cube roots of unity, prove that $(2 - ω + 2 ω^{2}) (2 + 2 ω - ω^{2}) = 9$

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Solution

$(2 - ω + 2 ω^{2}) (2 + 2 ω - ω^{2}) = 9$
Take L.H.S
$= (2 - ω + 2 ω^{2}) (2 + 2 ω - ω^{2})$
We know $1 + ω + ω^{2} = 0, ω^{3} = 1$
$= [2 (1 + ω^{2}) - ω] - [2 (1 + ω) - ω^{2}]$
$= [- 2 ω - ω] [- 2 ω^{2} - ω^{2}]$
$= (3 ω) (3 ω^{2})$
$= 9 ω^{2} = 9$
Hence proved.

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