If $\alpha ,\beta ,\gamma$ are cube roots of p( $p<0$ )then for any $x,y,z$,$\frac{{\alpha }^2{x}^2+{\beta }^2{y}^2+{\gamma }^2{z}^2}{{\beta }^2{x}^2+{\gamma }^2{y}^2+{\alpha }^2{z}^2}$is...

$\alpha ,\beta ,\gamma$ are the cube roots of $p<0$

$\therefore \alpha ,\beta ,\gamma =p^{1/3}=p,p\omega ,p{\omega }^2$

Let  $E=\frac{{\alpha }^2{x}^2+{\beta }^2{y}^2+{\gamma }^2{z}^2}{{\beta }^2{x}^2+{\gamma }^2{y}^2+{\alpha }^2{z}^2}$

$=\frac{p^2{x}^2+p^2{\omega }^2{y}^2+p^2\omega z^2}{p^2{\omega }^2{x}^2+p^2\omega y^2+p^2{z}^2}$

$=\frac{x^2+{\omega }^2{y}^2+\omega z^2}{{\omega }^2{x}^2+\omega y^2+z^2}$

$=\omega \left(\frac{x^2+{\omega }^2{y}^2+\omega z^2}{{\omega }^3{x}^2+{\omega }^2{y}^2+\omega z^2}\right)=\omega$