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

$t_{r+1}={}^{100}C{ }_r{\left(\sqrt[8]{5}\right)}^{100-r}.{\left(\sqrt[6]{2}\right)}^r$ 

 As 2 and 5 are coprime, $t_{r+1}$  will be rational if $100-r$  is a multiple of 8 and $r$  is a multiple of 6. also $0\le r\le 100$ 

. $\therefore r=0,6,12,\mathrm{.....},96$  $\Rightarrow 100-r=4,10,16,\mathrm{....},100$  But $100-r$  is to be a multiple of 8. So, $100-r=0,8,16,24,\mathrm{...96.}$ 

 The common terms in (1) and (2) are $16,40,64$  and $88$ . $\therefore r=84,60,36,12$  give rational terms.

 $\therefore$ The number of irrational terms $=101-4=97$ .