Answer
If the FOIL (first, outside, inside, last) method of multiplying the denominator by its conjugate is used, the outside and inside terms will cancel each other out.
Work Step by Step
Example:
$\frac{1+\sqrt x}{2+\sqrt x}$
$\frac{1+\sqrt x}{2+\sqrt x}*\frac{2-\sqrt x}{2-\sqrt x}$
$\frac{1*2+1*-\sqrt x + \sqrt x *2 + \sqrt x * -\sqrt x}{2*2+2*-\sqrt x + \sqrt x *2 + \sqrt x * -\sqrt x}$
$\frac{2-\sqrt x+2\sqrt x -x^2}{4-2\sqrt x +2\sqrt x -x^2}$
$\frac{2+\sqrt x-x^2}{4-x^2}$
$\frac{-x^2+2+\sqrt x}{-x^2+4}$
$\frac{x^2-2-\sqrt x}{x^2-4}$
