Find the cofficient of the term independent of x in the expansion of `((x+1)/(x^(2/3)-x^(1/3)+1)-(x-1)/(x-x^1/2))^10`

Answer

210

Explanation

`(x+1)/((x^(2//3) - x^(1//3))+1) - (x-1)/(x-x^(1//2))`
`= ((x^(1//3) +)(x^(2//3) - x^(1//3)+1))/(x^(2//3) - x^(1//3) + 1) - (x^(1//2) + 1)/(x^(1//2))`
`= x^(1//3) + 1 - 1 - x^(-1//2)`
` = x^(1//3) - x^(-1//2)`
`:. ((x+1)/(x^(2//3) - x^(1//3) + 1) - (x-1)/(x-x^(1//2)))^(10) = (x^(1//3) - x^(-1//2))^(10)`
Let `T_(r+1) = .^(100)C_(r) (x^(1//3))^(10-r) (-1)^(r) (-x^(-1//2))^(r )`
For the term independent of x,
`(10-r)/(3) - (r )/(2) = 0`
`rArr 20 - 2r - 3r = 0`
`rArr r = 4`
So, required coefficient `= .^(10)C_(4)(-1)^(4) = 210`