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If $z$ is a non real root of $\sqrt[7]{- 1}$ then $z^{86} + z^{175} + z^{289} =$

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$- 1$

answer is A.

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Detailed Solution

$Z = \sqrt[7]{- 1} \Rightarrow Z^{7} = - 1$

$∴ Z^{86} = Z^{84} . Z^{2} = {(Z^{7})}^{12} . Z^{2} = {(- 1)}^{12} . Z^{2} = Z^{2}$

$∴ Z^{175} = {(Z^{7})}^{25} = {(- 1)}^{25} = - 1$

$∴ Z^{289} = Z^{287} . Z^{2} = {(Z^{7})}^{41}, Z^{2} = {(- 1)}^{41} . Z^{2} = - Z^{2}$

$∴ Z^{86} + Z^{175} + Z^{289} = Z^{2} - 1 - Z^{2} = - 1$

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