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  4. If α , β are complex cube roots of unity, then the value of α2β2 + α10β2 + α2 β10 equals

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Q. If $\alpha , \beta$ are complex cube roots of unity, then the value of $\alpha^2\beta^2 + \alpha^{10}\beta^2 + \alpha^2 \beta^{10}$ equals

Complex Numbers and Quadratic Equations

Solution:

As $\alpha, \beta$ are complex cube roots of unity.
$\therefore \alpha = \omega, \beta = \omega^2$ or $\alpha = \omega^2, \beta = \omega$
$\therefore \alpha^2\beta^2 + \alpha^{10}\beta^2 + \alpha^2\beta^{10} = \omega^2(\omega^2)^2 + \omega^{10}\omega^4 + \omega^2\omega^{20}$
$= \omega^6 + \omega^{14} + \omega^{22}$
$= 1 + \omega^2 + (\omega^3)^7. \omega$
$= 1 + \omega^2 + \omega = 0 \,\,(\because \omega^3 = 1)$