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Properties of nth Root of a Complex Number

If z2-z+1=0...

Question

If $z^{2} - z + 1 = 0$ then $z^{n} - z^{- n}$ , where $n$ is a multiple of $3$ , is

2 (- 1)^{n}

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0

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(- 1)^{n + 1}

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N o n e o f t h e s e

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Solution

The correct option is B $0$
The roots are $- ω, - ω^{2}$
Now $ω^{3 n} = 1$
$- ω^{3 N} - ω^{- 3 N}$ , where $3 N = n$
$= 1 - 1$
$= 0$

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