Question

The number of rational terms in the binomial expansion of ${(4^{\frac{1}{4}}+5^{\frac{1}{6}})}^{120}$ is

Answer 1

$T_{r+1}={}^{120}{C}_r{\left(4^{1/4}\right)}^{120-r}\cdot {\left(5^{1/6}\right)}^r$
$={}^{120}{C}_r\cdot 2^{(60-r/2)}\cdot 5^{r/6}$
For $T_{r+1}$ to be rational, $r$ must be divisible by $6$.
$r=0,6,12,\dots ,120$
Number of rational terms $=21$