If $\alpha (\ne 1)$ is a fifth root of unity and $\beta (\ne 1)$ is a fourth root of unity then

$z=(1+\alpha )(1+\beta )\left(1+{\alpha }^2\right)\left(1+{\beta }^2\right)\left(1+{\alpha }^3\right)\left(1+{\beta }^3\right)$ equals

As $\beta \ne 1$ is a fourth root of unity,

${\beta }^4=1\Rightarrow (1-\beta )\left(1+\beta +{\beta }^2+{\beta }^3\right)=0$

As            $\beta \ne 1,1+\beta +{\beta }^2(1+\beta )=0$

$\begin{array}{l}\Rightarrow \hspace{0.25em} \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(1+\beta )\left(1+{\beta }^2\right)=0\\ \therefore \hspace{0.25em} \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}z=0\end{array}$