If $1, \omega , {\omega }^2$ are the cube roots of unity, then find the value of ${\left(1-\omega ,+{\omega }^2\right)}^5+{\left(1+\omega , -{\omega }^2\right)}^5$

Given $1,\omega ,{\omega }^2$ are the cube roots of unity

$$\begin{gathered} we know that 1+\omega +{\omega }^2=0 and {\omega }^3=1. \\ Now given ={\left(1-\omega +{\omega }^2\right)}^5+{\left(1+\omega -{\omega }^2\right)}^5 \\ ={\left(-\omega -\omega \right)}^5+{\left(-{\omega }^2-{\omega }^2\right)}^5 \\ ={\left(-2\omega \right)}^5+{\left(-2{\omega }^2\right)}^5={\left(-2\right)}^5\left\{{\omega }^5+{\omega }^{10}\right\} \\ =-32\left\{{\omega }^2+\omega \right\}=32\left(-1\right) \\ =32 \end{gathered}$$