The sum of rational terms in the expansion of  ${(\sqrt{2}+3^{1/5})}^{10}$ is

(r + 1)th term in the given expansion is given by ${\mathrm{t}}_{\mathrm{r}+1}{=}^{10}{\mathrm{C}}_{\mathrm{r}} 2^{\frac{10-\mathrm{r}}{2}} 3^{\frac{\mathrm{r}}{5}}$
where r = 0, 1, 2, ..., 10
For rational terms
r = a multiple of 5 = 0, 5, 10……..(1)
10 – r = a multiple of 2 = 0, 2, 4, 6, 8, 10 ……...(2)
From Eq. (1) and (2) possible values of r are : 0 and 10
∴ sum of rational terms

$\begin{array}{rl} & ={\mathrm{t}}_1+{\mathrm{t}}_{11}{=}^{10}{\mathrm{C}}_0\left(\sqrt{2}{)}^{10}{\left(3^{1/5}\right)}^0{+}^{10}{\mathrm{C}}_{10}\right(\sqrt{2}{)}^0{\left(3^{1/5}\right)}^{10}\\ & =2^5+3^2=32+9=41\end{array}$