The number of rational terms in the binomial expansion of 4 1 4 + 5 1 6 120 is_______.

We have,

${\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)}^{120}$

$T_{r+1}={}^{120}C_r{\left(4^{\frac{1}{4}}\right)}^{120-r}{\left(5\right)}^{\frac{r}{6}}$

$\Rightarrow T_{r+1}={}^{120}C_r{\left(2\right)}^{\frac{120-r}{2}}{\left(5\right)}^{\frac{r}{6}}$
 

For rational terms $r=6\lambda , 0\le r\le 120$, hence

$r=0, 6, 12, ..., 120$

So, total number of rational terms are $21$.