Cube Roots of Unity

Evaluate :  ${\left(-\frac{1}{2}+\frac{\sqrt{3}i}{2}\right)}^{1000}$.

If $x+\frac{1}{x}=1$, $\lambda =x^{4000}+\frac{1}{x^{4000}}$ and $\mu$ be the digit at the unit place of the number $2^{2^n}+1,$ where $n\in N$ then value of $\lambda +\mu$ is equal to

If $z=\frac{1}{2}\left(\sqrt{3}-i\right)$ and the least positive integral value of $n$ such that ${\left(z^{101}+i^{109}\right)}^{106}=z^n$ is $k$, then the value of $\frac{2}{5}k$ is equal to

If $\alpha , \beta$ and $\gamma$ are the roots of the equation $x^3-3x^2+3x+7=0,$ and $w$ is cube root of unity, then the value of $\frac{\alpha -1}{\beta -1}+\frac{\beta -1}{\gamma -1}+\frac{\gamma -1}{\alpha -1}$ is equal to

If ω is the cube root of unity, then value of the ${\left(1+\mathrm{\omega }-{\mathrm{\omega }}^2\right)}^2+{\left(1-\mathrm{\omega }+{\mathrm{\omega }}^2\right)}^2+1$ is

If $1, w, w^2$ are the cube roots of unity and if $\alpha =w+2w^2-3 then {\alpha }^3+12{\alpha }^2+48\alpha +3=$

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

If $z=\frac{\sqrt{3}+i}{2}$ , then ${\left(z^{101}+i^{103}\right)}^{105}$ is equal to

If $i= \sqrt{-1}$ then $4+5 {\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)}^{334}-3{\left(\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)}^{365}$ is equal to-

If $\mathrm{\omega }$ is a cube root of unity, then the value of ${\left(1\mathrm{ }+\mathrm{\omega }\right)}^3-{\left(1\mathrm{ }+{\mathrm{\omega }}^2\right)}^3$ is

If $1,\mathrm{ }\mathrm{\omega },\mathrm{ }{\mathrm{\omega }}^2$ are the cube root of unity, then ${\left(1\mathrm{ }+\mathrm{\omega }\right)}^3-{\left(1\mathrm{ }+{\mathrm{\omega }}^2\right)}^3$ is equal to -

If $\mathrm{\omega }$ is a cube root of unity, then ${\left(3+5\mathrm{\omega }+3{\mathrm{\omega }}^2\right)}^2+{\left(3+3\mathrm{\omega }+5{\mathrm{\omega }}^2\right)}^2$ is equal to -

${\left(-8\right)}^{1/3}$ is equal to

If $z+z^{-1}=1$ , then $z^{50}+z^{-50}$ is equal to 

If $\mathrm{\alpha }$ is a complex number such that ${\mathrm{\alpha }}^2+\mathrm{\alpha }+1=0$ , then ${\mathrm{\alpha }}^{31}$ is equal to

The value of ${\left(-1+\sqrt{-3}\right)}^{62}+{\left(-1-\sqrt{-3}\right)}^{62}$ is :

If $3^{49}\mathrm{ }\left(\mathrm{x}+\mathrm{i}\mathrm{y}\right)={\left(\frac{3}{2}+\frac{\sqrt{3}}{2}\mathrm{i}\right)}^{100}$ and $\mathrm{x}=\mathrm{k}\mathrm{y}$ , then $\mathrm{k}$ is -

$\sqrt{-1-\sqrt{1-\sqrt{1-...\infty }} }$ is equal to

If $\omega$ is a complex root of the equation $z^3=1,$ then $\omega +{\omega }^{\left(\frac{1}{2}+\frac{3}{8}+\frac{9}{32}+\frac{27}{128}+...\right)}$ is equal to

If $\omega =\frac{-1+\sqrt{3}i}{2}$, then ${\left(3+\omega +3{\omega }^2\right)}^4$ is