QJEEM24Ma203a: JEE Main 2024 Mathematics - Class 11 JEEM24Ma203a Solution: The sum of all rational terms in the expansion of is equal to :

$\begin{aligned}& T_{r+1}={ }^{15} \mathrm{C}_r\left(2^{1 / 5}\right)^{15-r}\left(5^{1 / 3}\right)^r \\& ={ }^{15} C_r 5^{r / 3} 2^{\left(3-\frac{r}{5}\right)}\end{aligned}$

For rational terms,

$\frac{r}{3}$ and $\frac{r}{5}$ must be integer

3 and 5 divide $r \Rightarrow 15$ divides $r \Rightarrow r=0$ and $r=15$

${ }^{15} C_0 5^0 2^3+{ }^{15} C_{15} 5^5 2^{(0)}$

$\begin{aligned}& =8+3125 \\& =3133\end{aligned}$