Question

Find the total number of irrational terms in the binomial expansion of $(\sqrt[4]{5}+\sqrt[5]{4})^{100}$.

Answer 1

$T _{r+1}={ }^{100} C _{r}(\sqrt[4]{5})^{100-r}(\sqrt[5]{4})^{r}$
$={ }^{100} C _{r}(5)^{\frac{100-r}{4}}(4)^{\frac{r}{5}}$
$T _{r+1}$ will be rational when $\frac{100-r}{4}$ and $\frac{r}{5}$ are integers.
This is possible when $100-r$ is a multiple of $4$ and $r$ is a multiple of $5 .$
$\Rightarrow r=0,4,8, \ldots, 100$ and $r=0,5,10, \ldots, 100$
$\Rightarrow r=0,20,40,60,80,100$
$\Rightarrow$ Number of rational terms $=6$
$\Rightarrow$ Total number of irrational terms $=101-6$
$=95$