Let,

$S={\left(3^{\frac{1}{4}}+7^{\frac{1}{6}}\right)}^{144}$

General term of $S$ is given by

$T_{r+1}={}^{144}C_r \left\{{\left(3\right)}^{\frac{144-r}{4}}\right\}\left\{{\left(7\right)}^{\frac{r}{6}}\right\}$

Now, for the rational terms

$\left(\frac{144-r}{4}\right)\in W\left(\text{whole number}\right)$

For this $r$ must be the multiple of $4$ therefore

$r=0, 4, 8, ..., 140, 144$

And,

$\left(\frac{r}{6}\right)\in W\left(\text{whole number}\right)$

For this $r$ must be the multiple of $6$ therefore

$r=0, 6, 12, 18, 24, ....., 138, 144$

Now, common terms are

$0, 12, 24, ...., 144$

This is an AP. So,

$144=0+\left(n-1\right)12$

$\Rightarrow n=13$.