The number of terms which are free from radical signs in the expansion of ${\left(y^{1/5}+x^{1/10}\right)}^{55}$, is

The general term in the expansion of ${\left(y^{1/5}+x^{1/10}\right)}^{55}$

s given by 

$T_{r+1} {=}^{55}{C}_{r }{\left(y^{1/5}\right)}^{55-r}{\left(x^{1/10}\right)}^r{=}^{55}{C}_{ r}{y}^{11-r/5}{x}^{r/10}$

clearly, $T_{r+1}$ will be independent of radical signs if $r/5$ and

$r$/$10$are integers, where $0\le r\le 55$.

$r=0 ,10 ,20 ,30 ,40 ,50$

Hence, there are $6$`terms in the expansion of the

 given expression which are independent of radicals.