If $1, \omega, \omega^2$ are the three cube roots of unity,then $(3 + \omega^2 + \omega^4)^6 = $

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 $x = \cos \theta + i\sin \theta$ and $y = \cos \phi + i\sin \phi$,then ${x^m}{y^n} + {x^{-m}}{y^{-n}}$ is equal to

Let $i=\sqrt{-1}$. If $\frac{(-1+i \sqrt{3})^{21}}{(1-i)^{24}}+\frac{(1+i \sqrt{3})^{21}}{(1+i)^{24}}=k$,and $n =[| k |]$ be the greatest integral part of $| k |$. Then $\sum_{ j =0}^{ n +5}( j +5)^{2}-\sum_{ j =0}^{ n +5}( j +5)$ is equal to ........ .

If the cube roots of unity are $1, \omega, \omega^2$,then the roots of the equation $(x - 2)^3 + 27 = 0$ are

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