EXERCISE 5.6

Given that $1,\omega ,{\omega }^2$ are cube roots of unity, show that ${\left(1-\omega +{\omega }^2\right)}^5+{\left(1+\omega -{\omega }^2\right)}^5=32$.

Evaluate ${\omega }^{20}\text{ and }{\omega }^{-30}$.

Prove that ${\left(2+5\omega +2{\omega }^2\right)}^6=729$.

If $1,\omega ,{\omega }^2$ are three cube roots of unity, then prove that $\frac{p+q\omega +r{\omega }^2}{r+p\omega +q{\omega }^2}={\omega }^2$.

If $\alpha ,\beta$ are complex cube roots of unity other than $1$, show that ${\alpha }^4+{\beta }^4+{\alpha }^{-1}{\beta }^{-1}=0$.

Find the value of $(-1+\sqrt{3}i{)}^{100}+(-1-\sqrt{3}i{)}^{100}$.

Prove that ${\left(\frac{-1+i\sqrt{3}}{2}\right)}^n+{\left(\frac{-1-i\sqrt{3}}{2}\right)}^n$ is equal to $2$ if $n$ is a multiple of $3$.

Prove that ${\left(\frac{-1+i\sqrt{3}}{2}\right)}^n+{\left(\frac{-1-i\sqrt{3}}{2}\right)}^n$ is equal to $-1$ if $n$ is not a multiple of $3$.