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(A) The roots of the equation $x^2 - x + 1 = 0$ are $\omega$ and $\omega^2$, where $\omega$ is the complex cube root of unity.Let $\alpha = \omega$ an

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(A) The roots of the equation $x^2 - x + 1 = 0$ are $\omega$ and $\omega^2$, where $\omega$ is the complex cube root of unity.Let $\alpha = \omega$ and $\beta = \omega^2$.We know that $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.Substitute the values into the expression: $\alpha^{200} + \beta^{206} + 2 = (\omega)^{200} + (\omega^2)^{206} + 2$.Simplify the powers: $\omega^{200} = \omega^{3 	imes 66 + 2} = (\omega^3)^{66} \cdot \omega^2 = 1^{66} \cdot \omega^2 = \omega^2$.$\omega^{412} = \omega^{3 	imes 137 + 1} = (\omega^3)^{137} \cdot \omega = 1^{137} \cdot \omega = \omega$.Thus, the expression becomes: $\omega^2 + \omega + 2$.Since $1 + \omega + \omega^2 = 0$, we have $\omega + \omega^2 = -1$.Therefore, $-1 + 2 = 1$.

If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 - x + 1 = 0$, then the value of $\alpha^{200} + \beta^{206} + 2$ is equal to

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