The sum of all those terms which are rational numbers in the expansion of ${\left(2^{\frac{1}{3}}+3^{\frac{1}{4}}\right)}^{12}$ is:

$r^{\mathrm{th}}$ term in the expansion of ${\left(2^{\frac{1}{3}}+3^{\frac{1}{4}}\right)}^{12}$ is 

$T_{r+1}={}^{12}C_r{\left(2\right)}^{\frac{r}{3}}·{\left(3\right)}^{\frac{12-r}{4}}$

Since, $T_{r+1}$ will be rational number, when $r = 0,3,6,9,12$and $r = 0,4,8,12$

$\Rightarrow r=0,12$ for rational term of the expansion

Therefore, $T_1+T_{13}=1\times 3^3+1\times 2^4\times 1$

$=27+16$

$=43.$