TITLE: If $a,b$ are positive rational numbers and $\sqrt a+\sqrt b$ is rational, then both of $\sqrt a,\sqrt b$ are rational numbers
QUESTION [6 upvotes]: I'm trying to show that  

If $a,b$ are positive rational numbers and $\sqrt a+\sqrt b$ is rational, then both of $\sqrt a,\sqrt b$ are rational numbers.

I squared the number $\sqrt a+\sqrt b$ and found that $\sqrt {ab}$ is rational…

REPLY [7 votes]: Another possible approach is the following. Since $\sqrt{a}+\sqrt{b}$ is rational, $(\sqrt{a}+\sqrt{b})^2 = a+b+2\sqrt{ab}$ is rational, hence $\sqrt{ab}$ is rational, so $ab$ is the square of a rational number and $\sqrt{b}$ is a rational number times $\sqrt{a}$, say $q\sqrt{a}$. That gives $\sqrt{a}+\sqrt{b}=(1+q)\sqrt{a}$, hence $\sqrt{a}$ is rational and $a$ is the square of a rational number, so the same holds for $b$.