In the expansion of ${\left(\sqrt[5]{3}+\sqrt[7]{2}\right)}^{24}$, the rational term is

$$\begin{gathered} {\left(3^{1/5}+2^{1/7}\right)}^{24} \\ {}^{24}{\mathrm{C}}_{\mathrm{r}} {\left(3\right)}^{\frac{24-\mathrm{r}}{5}} {\left(2\right)}^{\frac{\mathrm{r}}{7}} \end{gathered}$$

For rational terms $\mathrm{r}=0, 7, 14, 21, \& \frac{24-\mathrm{r}}{5}$ be an integer.

r=14 is Possible.

$\therefore {\mathrm{T}}_{15}$ is the rational term