If the expansion of `(y^(1//2)+x^(1//3))^(54)`, the number of terms free from radical sign (number of rational terms) are independent where |x| and |y| have no common factor except 1 annd x,y`ne`Q, is
(1) 9
(2) 8
(3) 10
(4) 11.

Explanation

Answer (3)
The general term in the expansion of `(y^(1//2)+x^(1//3))^(54)` is
`=.^(54)C_(r)*(y^(1//2))^(54-r_*(x^(1//3))^(r)`
`=54C_(r)*y^((54-r)/(2))*x^((r)/(3))`
The term will be rational iff `21^(n-r) and 31r` simultaneous. This is possible only when r is multiple of 6.
hence, possible values of r are 0,6,12 . . . 54
Hence, required number of terms free from radical sign are=10