Question

Number of integral terms in the expansion of ${\left(\sqrt{5}+\sqrt[8]{7}\right)}^{1024}$ is

Answer 1

Answer: 129

We have, $T_{r=1}{=}^{1024}{C}_r{\left(5\right)}^{\frac{\left(1024-r\right)}{2}}{\left(7\right)}^{\frac{r}{8}}$
This is an integer, if $1024-r$ is an even number and $r$ is a multiple of $8$
i.e. when $r$ is a multiple of $8$ since $0\le r\le 1024$
The number of multiples of $8$ which lies between $0$ and $1024$ is
$1024=0+\left(n-1\right)8$
$\Rightarrow n-1=\frac{1024}{8}=128$
$\Rightarrow n=129$