Question

How many terms are free from radical signs in the expansion of \((x^{1/5} + y^{1/10})\)⁵⁵ .

Answer 1

The general term in the expansion of \((x^\frac{1}{5}+y^\frac{1}{10})^{55}\) is given by 

T_{r + 1} = ⁵⁵C_r \((x^\frac{1}{5})^{55-r}\) \((y^\frac{1}{10})^{r}\)

T_{r + 1} = ⁵⁵C_r x^11-r/5 \(y^\frac{r}{10}\)

Clearly, T_{r + 1} will be free from radical signs, if \(\frac{r}{5}\) and \(\frac{r}{10}\)are integers for 0 ≤ r ≤ 55. 

∴ r = 0, 10, 20, 30, 40, 50.

Hence, there are 6 terms in the expansion of \((x^\frac{1}{5}+y^\frac{1}{10})^{55}\) which are independent of radical sign.