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). ` `
General term in expansion (using binomial theorem) would be
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.