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

The general term $T_{r+1}$ in the expansion of

$({\left.5^{1/2}+7^{1/8}\right)}^{1024}$ is given by

$\begin{array}{l} T_{r+1}{=}^{1024}{C}_r{\left(5^{1/2}\right)}^{1024-r}{\left(7^{1/8}\right)}^r\\ \Rightarrow T_{r+1}{=}^{1024}{C}_r{5}^{512-r/2}{7}^{r/8}\\ \Rightarrow T_{r+1}=\left({ }^{1024}{C}_r{5}^{512-r}\right)\left(5^{r/2}{7}^{r/8}\right)\\ \Rightarrow T_{r+1}=\left\{{ }^{1024}{C}_r{5}^{512-r}\right\}{\left(5^4\times 7\right)}^{r/8}\end{array}$

clearly, $T_{r+1}$ will be an integer, if

$r/8$ is an integer such that $0\le r\le 1024$

$\Rightarrow$  $r$is a multiple of $8$lying satisfying $0\le r\le 1024$

$\Rightarrow r=0 ,8 ,16 ,\dots ,1024$

$\Rightarrow$  $r$can assume $129$values 

Hence, there are $129$ integral terms in the expansions of  

$({\left.5^{1/2}+7^{1/8}\right)}^{1024}$