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Q.
If $\alpha$ and $\beta$ are imaginary cube roots of unity, then the value of $\alpha^{4}+\beta^{28}+\frac{1}{\alpha \beta}$ is
Complex Numbers and Quadratic Equations
Solution:
Since $\alpha$ and $\beta$ are complex roots of unity,
we may write $\alpha=\omega, \beta=\omega^{2}$
Hence, $\alpha^{4}+\beta^{28}+\frac{1}{\alpha \beta}=\omega^{4}+\left(\omega^{2}\right)^{28}+\frac{1}{\omega \cdot \omega^{2}}$
$=\omega+\omega^{56}+1=\omega+\omega^{2}+1=0$