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Find the number of terms which are free ...

Find the number of terms which are free from radical signs in the expansion of `(x^(1//5)+y^(1//10))^(55)dot`

Text Solution

Verified by Experts

In the expansion of `(y^(1//5)+x^(1//10))^(55)`,
`T_(r+1) = .^(55)C_(r)(y^(1//5))^(55-r)(x^(1//10))^(r)=.^(55)C_(r)y^(11-r//5)x^(r//10)`
Thus, `T_(r+1)` will be independent of radicals if the exponents `r//5` and `r//10` are integers for `0 le r le 55`, which is possible only when `r = 0`, `10, 20, 30 , 40, 50`.
Therefore, there are six terms, i.e., `T_(1), T_(11), T_(21), T_(31), T_(41), T_(51)` which are independent of radicals.
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