If $w$ is a complex cube root of unity, then the value of $w^{10} - w^7 + w^5 - w^2 + 1$ is:

Let $\alpha$ and $\beta$ be the roots of $x^2 - \sqrt{2}x + 1 = 0$. Then the value of $\alpha^{50} + \beta^{50}$ is:

If $1, \alpha_1, \alpha_2, \ldots, \alpha_{n-1}$ are the $n^{\text{th}}$ roots of unity,then $\sum_{1 \leq i < j \leq n-1} \alpha_i \alpha_j =$

$\left(\frac{1+\cos (3 \theta)+i \sin (3 \theta)}{1+\cos (3 \theta)-i \sin (3 \theta)}\right)^{20} = ?$

The product of all the values of $(\sqrt{3}-i)^{2/5}$ is

The general value of the real angle $\theta$, which satisfies the equation $(\cos \theta + i \sin \theta)(\cos 2\theta + i \sin 2\theta) \dots (\cos n\theta + i \sin n\theta) = 1$ is given by (assuming $k$ is an integer):