Cube Roots of Unity

Let  $\omega$ be a complex cube root of unity with  $\omega \ne 1\mathrm{.}$  A fair die is thrown three times.  $r_1,r_2\mathrm{and}{r}_3$  are the numbers obtained on the die, then the probability that  ${\omega }^{r_1}+{\omega }^{r_2}+{\omega }^{r_3}=0$ is

If $\omega$ is an imaginary cube root of unity and  ${(1+\omega )}^7=A+B\omega$, then real numbers $A$ and $B$ are respectively ­–

If $\omega$ is a cube root of unity, then the value of polynomial $\left|\begin{array}{ccc}x+1& \omega & {\omega }^2\\ \omega & x+{\omega }^2& 1\\ {\omega }^2& 1& x+\omega \end{array}\right|$ is

If $\omega$ is a cube root of unity, then $1+{\omega }^2=$

If $z^2+z+1=0$, where $z$ is a complex number, find the value of  ${\left(z+\frac{1}{z}\right)}^2+{\left(z^2+\frac{1}{z^2}\right)}^2+{\left(z^3+\frac{1}{z^3}\right)}^2+{\left(z^4+\frac{1}{z^4}\right)}^2+{\left(z^5+\frac{1}{z^5}\right)}^2+{\left(z^6+\frac{1}{z^6}\right)}^2$.

If $1,\omega ,{\omega }^2$ are the cube roots of unity, then find the value of $k$ if  $(a+2b{)}^2+{\left(a{\omega }^2+2b\omega \right)}^2+{\left(a\omega +2b{\omega }^2\right)}^2=kab$.

If $1,\omega ,{\omega }^2$ are the cube roots of unity, then find the value of $(2-\omega )\left(2-{\omega }^2\right)\left(2-{\omega }^{10}\right)\left(2-{\omega }^{11}\right)$ is 

Prove that the expression $x^{3p}+x^{3q+1}+x^{3r+2}$ where $p, q, r$ are integers, is divisible by $x^2+x+1$.

Factorise $a^2-ab+b^2$

Factorise $x^2+xy+y^2$

If $x=a+b, y=a\omega +b{\omega }^2$ and $z=a{\omega }^2+b\omega$, where $\omega$ is complex cube root of unity, show that $xyz=a^3+b^3$.

Prove that ${\left(\frac{i+\sqrt{3}}{-i+\sqrt{3}}\right)}^{200}+{\left(\frac{i-\sqrt{3}}{i+\sqrt{3}}\right)}^{200}=-1$

If $\alpha , \beta$ are two imaginary cube roots of unity, show that ${\alpha }^4{\beta }^4-\frac{1}{\alpha \beta }=0$

If $\omega$ is an imaginary cube roots of unity, then show that, $\left(1-\omega +{\omega }^2\right)\left(1-{\omega }^2+{\omega }^4\right)\left(1-{\omega }^4+{\omega }^8\right)\left(1-{\omega }^8+{\omega }^{16}\right)=16.$

If $\omega$ is an imaginary cube root of unity, show that, ${\left(1-\omega +{\omega }^2\right)}^5+{\left(1+\omega -{\omega }^2\right)}^5=32$

If $\omega$ is an imaginary cube roots of unity, then show that, $(1+\omega )\left(1+{\omega }^2\right)\left(1+{\omega }^4\right)\left(1+{\omega }^8\right)=1.$

If $\omega$ is an imaginary cube root of unity, show that, $\left(1-{\omega }^2\right)\left(1-{\omega }^4\right)\left(1-{\omega }^8\right)\left(1-{\omega }^{10}\right)=9$

Evaluate: $2 x^4+3 x^3+3 x^2-x+6$, when $x=\frac{1}{2}\left(1+i\sqrt{3}\right)$

Find the value of ${\left(i+\sqrt{3}\right)}^{100}+{\left(i-\sqrt{3}\right)}^{100}+2^{100}$.