Question

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

Answer 1

Answer: 95

$T_{r+1}={}^{100}{C}_r(\sqrt[4]{5}{)}^{100-r}(\sqrt[5]{4}{)}^r$
$={}^{100}{C}_r(5{)}^{\frac{100-r}{4}}(4{)}^{\frac{r}{5}}$
$T_{r+1}$ 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 $5.$
$\Rightarrow r=0,4,8,\dots ,100$ and $r=0,5,10,\dots ,100$
$\Rightarrow r=0,20,40,60,80,100$
$\Rightarrow$ Number of rational terms $=6$
$\Rightarrow$ Total number of irrational terms $=101-6$
$=95$