The sum of two irrational square roots

Question

This is very similar to this question, but I was wondering if there was a simpler proof. In particular, a proof that would prove that $\sqrt{x}+\sqrt{y}$ is an irrational number if both $\sqrt{x}$ and $\sqrt{y}$ are irrational.

Answer 1

Hint: If $xy$ is not a perfect square, show that $(\sqrt x + \sqrt y )^2 = x + y + 2\sqrt{ xy}$ is irrational, and use the fact that and if $u^2$ is irrational then so is $u$.

If $xy = n^2$ then show that $\sqrt x + \sqrt y = \sqrt x\left(1+ \frac n {x}\right)$is irrational.

Answer 2

If $x$ and $y$ are integers then so is $x - y$ and further if $\sqrt x + \sqrt y$ is rational then $\sqrt x - \sqrt y = \frac{x - y}{\sqrt{x} + \sqrt{y}}$ is also rational. Then by sums and differences so are both $\sqrt{x}$ and $\sqrt{y}$. Therefore if we assume that at least one of $\sqrt{x}$ and $\sqrt{y}$ are irrational than so is their sum.