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(C) Since $\alpha$ and $\beta$ are the imaginary cube roots of unity,we can write $\alpha = \omega$ and $\beta = \omega^2$ (or vice versa).Given expre

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(C) Since $\alpha$ and $\beta$ are the imaginary cube roots of unity,we can write $\alpha = \omega$ and $\beta = \omega^2$ (or vice versa).Given expression: $\alpha^4 + \beta^{28} + \frac{1}{\alpha\beta}$.Substituting the values: $\omega^4 + (\omega^2)^{28} + \frac{1}{\omega \cdot \omega^2}$.Using the properties $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$:$\omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega$.$\beta^{28} = (\omega^2)^{28} = \omega^{56} = \omega^{54} \cdot \omega^2 = (\omega^3)^{18} \cdot \omega^2 = 1^{18} \cdot \omega^2 = \omega^2$.$\frac{1}{\alpha\beta} = \frac{1}{\omega \cdot \omega^2} = \frac{1}{\omega^3} = \frac{1}{1} = 1$.Therefore,$\alpha^4 + \beta^{28} + \frac{1}{\alpha\beta} = \omega + \omega^2 + 1 = 0$.

If $\alpha$ and $\beta$ are imaginary cube roots of unity,then the value of $\alpha^4 + \beta^{28} + \frac{1}{\alpha\beta}$ is:

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