$T_{r+1}={}^{60}C_r{\left(\frac{\sqrt{x}}{5^{{\displaystyle \frac{1}{4}}}}\right)}^{60-r}{\left(\frac{\sqrt{5}}{x^{{\displaystyle \frac{1}{3}}}}\right)}^r$

For $x^{10}$, $\frac{60-r}{2}-\frac{r}{3}=10\Rightarrow r=24$

Hence coefficient of $x^{10}$ will be ${}^{60}C_{24}{5}^3$

Now, exponent of $5$ in ${}^{60}C_{24}$ will be the power of $5$ in $\frac{60!}{24!36!} \text{i.e.} \frac{5^{14}}{5^4·5^8}=5^2$

i.e. $k=5$