The sum of all rational terms in the expansion of ${\left(3^{1/5}+2^{1/3}\right)}^{15}$ is

The general term in the expansion of ${\left(3^{1/5}+2^{1/3}\right)}^{15}$ is

$\begin{array}{c}{T}_{r+1}{=}^{15}{C}_r{\left(3^{1/5}\right)}^{15-r}\cdot {\left(2^{1/3}\right)}^r\\ {=}^{15}{C}_r\cdot 3^{3-\frac{r}{5}}\cdot 2^{\frac{r}{3}}\end{array}$

Now, $0\le r\le 15$

For $r = 0, 15$

Rational terms are $T_{0+1}$ and $T_{15+1}$

Then, $T_{0+1}{=}^{15}{C}_0\cdot 3^3\cdot 2^0=27$

and $T_{15+1}{=}^{15}{C}_{15}\cdot 3^0\cdot 2^5=32$

$\therefore$ Sum of all rational terms = 27 + 32 = 59