If x, y are both positive rational numbers, then $(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})$ is

1. a rational number
2. an irrational number
3. neither rational nor irrational number
4. both rational as well as irrational number

$(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y}) = \sqrt{x} × \sqrt{x} - \sqrt{x} × \sqrt{y} - \sqrt{x} × \sqrt{y} - \sqrt{y} × \sqrt{y} \\[1.5em] \Rightarrow (\sqrt{x})^2 - \sqrt{xy} + \sqrt{xy} -(\sqrt{y})^2 = x - y \\[1.5em]$

Since, x, y are both positive rational numbers, so the difference of two positive rational numbers is also a rational number .

Therefore, $x - y$ is also a rational number. Hence, $(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})$ is a rational number.

∴ Option 1, is the correct option.