If $\omega$  be non-real complex cube roots of unity, ${\alpha }^r(1\le r\le n-1)$ is a non real nth root of unity  then the value of $\frac{{\displaystyle \prod _{p=1}^4(\omega +{\alpha }^p)}}{{\displaystyle \prod _{q=1}^7({\omega }^2-{\alpha }^q)}}$  is equal to

$\frac{z^n-1}{z-1}={\displaystyle \prod _{r=1}^{n-1}(z-{\alpha }^r)}$  --- (1)

$n=5,z=-w,r=p$

$\Rightarrow \frac{-w^5-1}{-w-1}={\displaystyle \prod _{r=1}^4(-w-{\alpha }^r)}\Rightarrow {\displaystyle \prod _{p=1}^4(w+{\alpha }^p)}=\frac{1}{w}$  --- (2)

$z=w^2,n=8,r=q$ $\Rightarrow {\displaystyle \prod _{q=1}^7(w^2-{\alpha }^2)}=-\frac{1}{w^2}$  --- (3)

    $\frac{\left(2\right)}{\left(3\right)}\Rightarrow -w=-\left(\frac{-1+i\sqrt{3}}{2}\right)=\frac{1-i\sqrt{3}}{2}$