The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\rig

Question

The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}$ is

Professor PC Thomas Classes · Answer

(21) : General term in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}ight)^{120}$\ $=T_{r+1}={ }^{120} C_{r}\left(2^{1 / 2}ight)^{120-r}(5)^{r / 6}$ For rational terms, $r=6 k, r=120-2 k_{1}$ and $0 \leq r \leq 120$\ $\Rightarrow \quad r=6 k$ i.e., $r=0,6,12, \ldots \ldots . ., 120$\ So, total number of terms is 21 .\