If $a$ and $b$ are real numbers such that $(2+\alpha)^{4}=a+b \alpha,$ where $\alpha=\frac{-1+i \sqrt{3}}{2},$ then $a+b$ is equal to

Evaluate: $[\sqrt{2}(\cos 56^{\circ} 15^{\prime} + i \sin 56^{\circ} 15^{\prime})]^8$

If $n$ is an integer which leaves a remainder of $1$ when divided by $3$,then $(1+\sqrt{3}i)^n + (1-\sqrt{3}i)^n$ equals

If $i^2 = -1$,then $(1 + \sqrt{3} i)^{2022} - (\sqrt{3} - i)^{2022} = $

If $1, \omega, \omega^2$ are the cube roots of unity,then
$1(2+\frac{1}{\omega})(2+\frac{1}{\omega^2})+2(3+\frac{1}{\omega})(3+\frac{1}{\omega^2})+3(4+\frac{1}{\omega})(4+\frac{1}{\omega^2})+\ldots 10 \text{ terms} =$

If $\left(\frac{3}{2}+i \frac{\sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)$ where $x$ and $y$ are real,then the ordered pair $(x, y)$ is