If ${x_n} = \cos \left( \frac{\pi }{4^n} \right) + i \sin \left( \frac{\pi }{4^n} \right)$,then ${x_1} \cdot {x_2} \cdot {x_3} \dots \infty = $

If $\alpha, \beta$ are non-real cube roots of $2$,then $\alpha^6 + \beta^6$ equals

If $\omega$ is a complex cube root of unity,then $\omega^{\left(\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+\ldots \infty\right)}+\omega^{\left(\frac{1}{2}+\frac{3}{8}+\frac{9}{32}+\ldots \infty\right)}$ is equal to

Let $z = \frac{1 - i \sqrt{3}}{2}$,where $i = \sqrt{-1}$. Then the value of $21 + \left(z + \frac{1}{z}\right)^{3} + \left(z^{2} + \frac{1}{z^{2}}\right)^{3} + \left(z^{3} + \frac{1}{z^{3}}\right)^{3} + \dots + \left(z^{21} + \frac{1}{z^{21}}\right)^{3}$ is .... .

In the Argand plane,no value of $\sqrt[3]{1-i \sqrt{3}}$ lies in which quadrant?

$(-\sqrt{3} + i)^{53}$,where $i^2 = -1$,is equal to: