If there any way to expand the following?

$$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}}$$

and more generally, a way to expand

$$\left(\sum_{i=1}^nx_i\right)^{\frac{p}{q}}$$

where $gcd(p,q) = 1$

More closed to my original problem, is there any formula for: $$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}} - \sum_{i=1}^nx_i^{\frac{1}{2}}$$