Question

The number of irrational terms in the expansion of ${(\sqrt[8]{85}+\sqrt[6]{62})}^{100}$ is

Answer 1

We have, ${(\sqrt[8]{85}+\sqrt[6]{62})}^{100}$
Here, $T_{r+1}={}^{100}{C}_r{\left(\sqrt[8]{85}\right)}^{100-r}{\left(\sqrt[6]{62}\right)}^r$
$={}^{100}{C}_r\left(5{)}^{\frac{100-r}{8}}\right(2{)}^{\frac{r}{6}}$

Thus, for $T_{r+1}$ to be rational
$\left(1\right)$ $100-r$ should be a multiple of $8$, and
$\left(2\right)$ $r$ should be a multiple of $6$

So possible values of $r(0\le r\le 100)$ for terms to be rational are $\{12,36,60,84\}$

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