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Properties of nth Root of a Complex Number

If α is a non...

Question

If $α$ is a non real root of $z = (1)^{1 / 5},$ then the value of $z^{(1 + α + α^{2} + α^{- 2} + α^{- 1})}$ is

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Solution

$1 + α + α^{2} + α^{- 2} + α^{- 1}$
$= 1 + α + α^{2} + α^{3} + α^{4} (∵ α^{5} = 1)$
$= 0 (∵ sum of roots is zero)$
$∴ z^{(1 + α + α^{2} + α^{- 2} + α^{- 1})} = z^{0} = 1$

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Properties of nth Root of a Complex Number

Standard XII Mathematics

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