The product of the distinct $(2n)^{\text{th}}$ roots of $1+i\sqrt{3}$ is equal to:

If $1, \omega, \omega^{2}$ are the cube roots of unity,then $(1+\omega)(1+\omega^{2})(1+\omega^{4})(1+\omega^{8})$ is equal to

If $\omega$ is a complex cube root of unity, then the value of the expression $2(1 + \frac{1}{\omega})(1 + \frac{1}{\omega^2}) + 3(2 + \frac{1}{\omega})(2 + \frac{1}{\omega^2}) + ... + (n + 1)(n + \frac{1}{\omega})(n + \frac{1}{\omega^2})$ is...

If $\alpha$ is an imaginary cube root of unity,then for $n \in N$,the value of $\alpha^{3n + 1} + \alpha^{3n + 3} + \alpha^{3n + 5}$ is

Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{2} x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to

If ${x_r} = \cos \left( \frac{\pi }{2^r} \right) + i\sin \left( \frac{\pi }{2^r} \right)$,then the product ${x_1} \cdot {x_2} \cdot {x_3} \cdots \infty$ is: