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

${\left(\sqrt[5]{3}+\sqrt[7]{2}\right)}^{24}$

General term $={}^{24}{\mathrm{C}}_{\mathrm{r}} {\left(3^{\frac{1}{5}}\right)}^{24-\mathrm{r}} {\left(2^{\frac{1}{7}}\right)}^{\mathrm{r}}$

                        $={}^{24}{\mathrm{C}}_{\mathrm{r}} 3\frac{24-\mathrm{r}}{5}·2^{\frac{\mathrm{r}}{7}}$

For rational term exponents should be Integers $\Rightarrow \frac{\mathrm{r}}{7}, \frac{24-\mathrm{r}}{5}$ be Integers.

$\mathrm{r}=0, 7, 14, 21$

if $\mathrm{r}=14, \frac{24-\mathrm{r}}{5}$ is also an integer 

$\therefore {\mathrm{T}}_{\mathrm{r}+1}={\mathrm{T}}_{15}$ is the rational term.