Is it possible for integer square roots to add up to another? I initially was wondering if it were possible for there to be three $x,y,z \in \mathbb{Q}$ and $\sqrt{x},\sqrt{y},\sqrt{z} \notin \mathbb{Q}$ such that $\sqrt{x} + \sqrt{y} = \sqrt{z}$. I had suspected not, but then I found $x = \dfrac{1}{2}, y = \dfrac{1}{2}$ and thus $\sqrt{x} + \sqrt{y} = \sqrt{2}$.
I suspect there are no integer solutions where the numbers are not all square, but I couldn't prove it. Nonetheless I figured I'd ask if it were possible when $x, y, z \in \mathbb{N}$ and $\sqrt{x},\sqrt{y},\sqrt{z} \notin \mathbb{N}$ that $\sqrt{x} + \sqrt{y} = \sqrt{z}$? And if not, can someone prove it?

Some examples for each variable $x$ and $y$ up to 100:

