The expansion of `(2x^(2)-1/sqrt(x))^(n)` where n is a positive integer,has a term that is independent of x.Find the smallest value of n.Thanks!   (Using the binomial theorem if possible). ` `

Expert Answers
tiburtius eNotes educator| Certified Educator

General term in expansion (using binomial theorem) would be

`(-1)^k((n),(k))(2x^2)^(n-k)(1/(sqrt x))^k`

However, we are interested only in term that is independent of `x` so we will disregard everything else. If the term is independent of `x` it means that both powers of `x` must be the same i.e.

`x^(2(n-k))=(sqrt x )^k`

`x^(2(n-k))=x^(k/2)`    because `sqrt x=x^(1/2)`






In order to find the smallest `n` we need to determine the smallest `k>0` for which `n` is a whole number. Obviously we get that for `k=4` (for `k=1,2,3`  `n` is not a whole number). 



The smallest value of `n` is 5.