Question

The number of rational terms in the expansion of the ${(7^{\frac{1}{9}}+{13}^{\frac{1}{11}})}^{1980}$ is

Answer 1

The general term of the expansion of the ${(7^{\frac{1}{9}}+{13}^{\frac{1}{11}})}^{1980}$ is given by $T_{r+1}={}^{1980}{C}_r{\left(7^{\frac{1}{9}}\right)}^{1980-r}{\left({13}^{\frac{1}{11}}\right)}^r\Rightarrow T_{r+1}={}^{1980}{C}_r\left(7{)}^{220}\right(7{)}^{-\frac{r}{9}}(13{)}^{\frac{r}{11}}$
So, for the rational terms $r$ must be multiple of both $9\&11$. In other words, $r$ must be multiple of $99$.
$r=0,99,198,....,1980$, total $21$ terms