$$ \begin{aligned} & \text { General term of the expansion }\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680} \\\\ & \qquad={ }^{680} C_r\left(3^{1 / 2}\right)^{680-r}\left(5^{1 / 4}\right)^r={ }^{680} C_r \times 3^{\frac{680-r}{2}} \times 5^{\frac{r}{4}} \end{aligned} $$

The term will be integral if $r$ is a multiple of 4 .

$$ \begin{gathered} \therefore r=0,4,8,12, \ldots, 680(\text { which is an } \mathrm{AP}) \\\\ 680=0+(n-1) 4 \\\\ n=\frac{680}{4}+1=171 \end{gathered} $$