find the term independent of `x` in the expansion of `(sqrtx/sqrt3+sqrt3/(2x^2))^10`

Explanation

Let `T_(r+1)` be independent of x.
Here ,`T_(r+1) = .^(10) C_(r) xx (sqrt(x/3))^((10-r)) xx ((sqrt(3))/(2x^(2)))^(r)`
`T_(r+1) = .^(10) C _(r) xx (x/3)^(((10-r))/2) xx 3^(r/2) xx 1/(2^(r) x^(2r))`
`=.^(10)C_(r) xx x^((5-r/2 -2r)) xx1/(3^((5-r/2))) xx 3^(r/2) xx 2^(-r)`
`=.^(10) C_(r) xx x^((5-(5r)/2)) xx3^((r-5)) xx 2 ^(-r)." "`...(i)
Now, `T_(r+1)` will be independent of x only when the power of x in it is 0.
`:. 5- (5r)/2 = 0 rArr (5r)/2 rArr =5 rArr r = 2 rArr r + 1 = 3 .`
Thus, `T_(3)` is independent of x.
Putting `r=2` in (i), we get
`T_(3) = T _(2+1)`
`=.^(10) C _(2) xx 3^((2-5)) xx 2 ^(-2) xx x^(0) = ((10 xx9)/(2 xx 1) xx 3^(-3) xx 2 ^(-2)).`
`= (45/(3^(3) xx 2^(2))) = ( 45/(27 xx 4)) = 5 /12.`
Hence, the term independent of x in the given expansion is `5/12`.