Question

Number of rational terms in the expansion of ${(\sqrt{2}+\sqrt[5]{54})}^{100}$ is

Answer 1

General term in the expansion of ${(\sqrt{2}+\sqrt[5]{54})}^{100}$ will be
$T_{r+1}={}^{100}{C}_r\left(2{)}^{(100-r)/2}\right(4{)}^{r/5}\Rightarrow T_{r+1}={}^{100}{C}_r(2{)}^{\frac{(500-r)}{10}}$
Now for the rational term $\frac{500-r}{10}$ has to be an integer
$\therefore r=0,10,20,\cdots ,100$
Hence there will be $11$ rational terms