. In the expansion of ${\left(5^{1/2}+7^{1/8}\right)}^{1024}$,the number of integral terms is

${\mathrm{T}}_{\mathrm{r}+1}{=}^{1024}{\mathrm{C}}_{\mathrm{r}}{\left(5^{1/2}\right)}^{1024-\mathrm{r}}{\left(7^{1/8}\right)}^{\mathrm{r}}$

Now this term is an integer if 1024 - r is an even integer, for
which r= 0,2, 4, 6, ..., 1022, 1024 of which r =0, 8, 16,24,...,1024 are divisible by 8 which makes rl8 an integer.
For A.P ., r =0, 8, 16,24,..., 1024

$1024=0+(\mathrm{n}-1)8\text{ or }\mathrm{n}=129$