elementary number theory - The sum of square roots of non-perfect squares is never integer

My question looks quite obvious, but I'm looking for a strict proof for this:

Why can't the sum of two square roots of non-perfect squares be an integer?

For example: $\sqrt8+\sqrt{15}$ isn't an integer. Well, I know this looks obvious, but I can't prove it...

Answer

Assume that $a, b \in \mathbb N $ are not perfect squares and
$ \sqrt a + \sqrt b = n \in \mathbb N$. Then
$$
a = (n - \sqrt b)^2 = n^2 - 2 n \sqrt b + b

$$
which means that $\sqrt b$ is a rational number. This contradicts the fact
that the square root of an integer is either an integer or a irrational number.