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

We have, 

$\begin{array}{l} (\sqrt[8]{5}+\sqrt[6]{2}{)}^{100}=\sum _{r=0}^{100}{ }^{100}{C}_{r }\left(\sqrt[8]{5}{)}^{100-r}\right(\sqrt[6]{2}{)}^r\\ \Rightarrow (\sqrt[8]{5}+\sqrt[6]{2}{)}^{100}=\sum _{r=0}^{100}{ }^{100}{C}_r 5^{100-r}\frac{r}{8}=\sum _{r=0}^{100} T_{r+1}\end{array}$

         where $T_{r+1}{=}^{100}{C}_r{5}^{\frac{100-r}{8}}{2}^{\frac{r}{6}}$

   Clearly  $T_{r+1}$ will be an integer if $\frac{100-r}{8}$ and $\frac{r}{6}$ are integers.

 This is possible when $100-r$ is a multiple of $8$and $r$is a multiple 

 of $6$

$\Rightarrow 100-r=0,8,16,\dots .,96$ and $,r=0,6,12,\dots ,96$

$\Rightarrow r=4,12,20,\dots ,100$ and $r=0,6,12,\dots .,96$

$\Rightarrow r=12, 36, 60, 84$

Hence, there are $4$rational terms and $101 - 4 = 97$ irrational

terms.