(i) Find the coefficeint of `x^(-1)` in the expansion of `(1+x)^(n)(1+(1)/(x))^(n)`
(ii) Find the term inependent of x in the expansion of `(x^((2)/(3))+4x^((1)/(3))+4)^(5)[(1)/(x^((1)/(3))-1)+(1)/(x^((2)/(3))+x^((1)/(3))+1)]^(-9)`

Explanation

(i) The given expression is
`(1+x)^(n)(1+(1)/(x))^(n)=((1+x)^(2n))/(x^(n))`
`implies`coefficient of `(1)/(x)` in `(1+x)^(n)(1+(1)/(x))^(n)=`coefficient of `x^(n-1)` in `(1+x)^(2n)`
`=.^(2n)C_(n-1)=(2n!)/((n-1)!(n+1)!)`
(ii) we have,
`(x^(2//3)+4x^(1//3)+4)^(5)[(1)/(x^(1//3)-1)+(1)/(x^(2//3)+x^(1//3)+1)]^(-9)`
`=[(x^(1//3)+2)^(2)]^(5)[(x^(2//3)+x^(1//3)+1+x^(1//3)-1)/((x-1))]^(-9)`
`=(2+x^(1//3))^(10)*((x-1)^(9))/([x^(1//3)(x^(1//3)+2)]^(9))=((2+x^(1//3))^(10)*(x-1)^(9))/(x^(3)(2+x^(1//3))^(9))`

`=(-2+x^(1//3))(.^(9)C_(0)-.^(9)C_(1)x+^(9)C_(2)x^(2)-.^(9)C_(3)x^(3)+ . . . .))/(x^(3))`
Clearly the independent term in the given expansion
`=-2x-^(9)C_(3)=168`