JEE Main · 2020 · 03 Sep 2020 · Shift 1 · Binomial Theorem

A general term of ${\left(3^{1/2}+5^{1/8}\right)}^n$ is $\left(\genfrac{}{}{0ex}{}{n}{r}\right)3^{(n-r)/2}{5}^{r/8}$, and it is an integer only when $r$ is a multiple of $8$ (to clear the eighth root) and $n-r$ is even. The admissible values $r=0,8,16,\dots$ up to $n$ give the integral terms, and for there to be exactly $33$ of them the largest must be $r=8\times 32=256$. Taking $n=256$, which is even so that $n-r$ stays even throughout, gives the required $33$ terms with the least possible $n$.