how many terms are free from radical signs in the expansionof (x^1/5+y^1/10)⁵⁵,and what is meant by radical sign here?

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$$\begin{gathered} \mathrm{Sign} \mathrm{like} \sqrt[\mathrm{x}]{} \mathrm{before} \mathrm{any} \mathrm{number} \mathrm{is} \mathrm{called} \mathrm{radical} \mathrm{sign}. \\ \mathrm{e}.\mathrm{g}.=3^{\frac{1}{3}}= \sqrt[3]{3}, 7^{\frac{1}{5}}=\sqrt[5]{7} \\ \mathrm{Now} \mathrm{in} \sqrt[3]{2^6}={\left(2^6\right)}^{\frac{1}{3}}=2^2=4, \mathrm{hence} \mathrm{if} \mathrm{radical} \mathrm{sign} \mathrm{is} \mathrm{a} \mathrm{multiple} \mathrm{of} \\ \mathrm{power} \mathrm{the} \mathrm{number}, \mathrm{then} \mathrm{radical} \mathrm{sign} \mathrm{will} \mathrm{be} \mathrm{removed}. \\ \mathrm{We} \mathrm{have} \\ {\left({\mathrm{x}}^{\frac{1}{5}}+{\mathrm{y}}^{\frac{1}{10}}\right)}^{55} \\ \mathrm{General} \mathrm{term} \mathrm{will} \mathrm{be} \mathrm{given} \mathrm{by}: \\ {\mathrm{T}}_{\mathrm{r}+1}{=}^{55}{\mathrm{C}}_{\mathrm{r}}{\left(\mathrm{x}\right)}^{\left(\frac{55-\mathrm{r}}{5}\right)}.{\left(\mathrm{y}\right)}^{\frac{\mathrm{r}}{10}} \left[0\le \mathrm{r}\le 55\right] \\ \mathrm{To} \mathrm{get} \mathrm{radical} \mathrm{free} \mathrm{terms}, \mathrm{r} \mathrm{should} \mathrm{be} \mathrm{multilple} \mathrm{of} 10 \\ \mathrm{and} \left(55-\mathrm{r}\right) \mathrm{should} \mathrm{be} \mathrm{multiple} \mathrm{of} 5. \\ \mathrm{If} \mathrm{r} \mathrm{is} \mathrm{multilple} \mathrm{of} 10, \\ \mathrm{r}=\left\{0,10,20,30,40,50\right\} \left[\mathrm{As} 0\le \mathrm{r}\le 55\right] \\ \mathrm{If} \mathrm{r} \mathrm{is} \mathrm{multilple} \mathrm{of} \left(55-\mathrm{r}\right), \mathrm{then}: \\ \mathrm{r}=\left\{0,5,10,15,20,25,30,35,40,45,50,55\right\} \\ \mathrm{Taking} \mathrm{common} \mathrm{values} \mathrm{we} \mathrm{get}: \\ \mathrm{r}=\left\{0,10,20,30,40,50\right\}, \\ \mathrm{Hence} 6 \mathrm{terms} \mathrm{will} \mathrm{be} \mathrm{radical} \mathrm{free}. \end{gathered}$$

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