Question

Find the number of rational terms in the expansion of ${(9^{1/4}+8^{1/6})}^{1000}.$

Answer 1

The general term in the expansion of ${(9^{1/4}+8^{1/6})}^{1000}$ is
$T_{r+1}={}^{1000}{\text{C}}_r{\left(9^{\frac{1}{4}}\right)}^{1000-r}{\left(8^{\frac{1}{6}}\right)}^r={}^{1000}{\text{C}}_r{3}^{\frac{1000-r}{2}}{2}^{\frac{r}{2}}$
The above term will be rational if exponent of $3$ and $2$ are integers
It means $\frac{1000-r}{2}$ and $\frac{r}{2}$ must be integers
The possible set of values of $r$ is $\{0,2,4,...,1000\}$
Hence, number of rational terms is $501$.