The number of irrational terms in the expansion of$( \sqrt [ 8 ] { 5 } + \sqrt [ 6 ] { 2 } ) ^ { 100 } ,$ is

$\begin{array}{l}\text { We have, }(\sqrt[6]{5}+\sqrt[8]{2})^{100} \\ \text { Here, } T_{r+1}={ }^{100} C_{r}(\sqrt[8]{5})^{100-r}(\sqrt[6]{2})^{r} \\ ={ }^{100} C_{r}(5)^{\frac{100-r}{8}}(2)^{\frac{r}{6}}\end{array}$

Thus, for $T_{r+1}$ to be rational

(1) 100 - $r$ should be a multiple of 8 , and

(2) $r$ should be a multiple of 6

So possible values of $r(O \leq r \leq 100)$ for terms to be rational are $\{12,36,60,84\}$

$\therefore$ number of irrational terms $=$ total terms - number of rational terms $=$ $101-4=97$