In the expansion of $\left(3^{1 / 4}+7^{1 / 6}\right)^{144}$
General term is
$$
\begin{aligned}
T_{r+1} & ={ }^{144} C_r\left(3^{1 / 4}\right)^{144-r}\left(7^{1 / 6}\right)^r \\
& ={ }^{144} C_r(3)^{\frac{144-r}{4}} 7^{\frac{r}{6}}
\end{aligned}
$$
For rational terms $144-r$ is divisible of 4 .
$$
\therefore \quad r=0,4,8, \ldots
$$
and $r$ is divisible by 6
$$
\therefore \quad r=0,6,12, \ldots .
$$
So, common value of $r$ will be multiple of 12 .
$$
\begin{aligned}
& \therefore \quad i=0,12,24, \ldots 144 \\
& \text { total values of } r=13 \\
& \therefore \text { Number of rational terms }=13
\end{aligned}
$$