If $1,{\mathrm{\alpha }}_1,{\mathrm{\alpha }}_2,\dots \dots ,{\mathrm{\alpha }}_8$ are  ninth roots of unity (taken in counter-clockwise direction) then $\left|\left(2-{\mathrm{\alpha }}_1\right)\left(2-{\mathrm{\alpha }}_3\right)\left(2-{\mathrm{\alpha }}_5\right)\left(2-{\mathrm{\alpha }}_7\right)\right|$ is equal to

$\begin{array}{l}(\mathrm{z}-1)\left(\mathrm{z}-{\mathrm{\alpha }}_1\right)\left(\mathrm{z}-{\mathrm{\alpha }}_2\right)\dots \dots \left(\mathrm{z}-{\mathrm{\alpha }}_8\right)={\mathrm{z}}^9-1\\ \Rightarrow \left(2-{\mathrm{\alpha }}_1\right)\left(2-{\mathrm{\alpha }}_2\right)\dots \left(2-{\mathrm{\alpha }}_8)=2^9-1\right.\\ \Rightarrow \left|\left(2-{\mathrm{\alpha }}_1\right)\left(2-{\mathrm{\alpha }}_2\right)\dots \left(2-{\mathrm{\alpha }}_8)\right.\right|=512-1=511\\ \end{array}$
but ${\mathrm{\alpha }}_1,{\mathrm{\alpha }}_8;{\mathrm{\alpha }}_2,{\mathrm{\alpha }}_7;{\mathrm{\alpha }}_3,{\mathrm{\alpha }}_6;{\mathrm{\alpha }}_4,{\mathrm{\alpha }}_5$ are pair wise conjugate complex numbers.
$\therefore \left|\left(2-{\mathrm{\alpha }}_1\right)\left(2-{\mathrm{\alpha }}_3\right)\left(2-{\mathrm{\alpha }}_5\right)\left(2-{\mathrm{\alpha }}_7\right)\right|=\sqrt{511}$