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Properties of nth Root of a Complex Number

If the fourth...

Question

If the fourth roots of unity are $z_{1}, z_{2}, z_{3}, z_{4}$ then $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2}$ is equal to

1

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0

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i

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None of these

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Solution

The correct option is B $0$
If $z$ be the fourth roots of unity then, $z^{4} = 1$
$\Rightarrow (z^{4} - 1) = 0 \Rightarrow (z^{2} - 1) (z^{2} + 1) = 0$
$\Rightarrow z = \pm 1, \pm i,$ where $i^{2} = - 1$
$∴ z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2} = 1 + 1 + i^{2} + i^{2} = 2 - 2 = 0$

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z_{1,} z_{2,} z_{3,} z_{4,}

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| Z_{1} | = | Z_{2} | = | Z_{3} | = 1

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