Why So Rational?

Let $a + \sqrt{ ab } = X, b + \sqrt{ ab } = Y$ .
$X + Y = a + 2 \sqrt{ab} + b = ( \sqrt{ a } + \sqrt{ b} ) ^ 2$ , so $\sqrt{ X + Y } = \sqrt{a} + \sqrt{b}$ .
$\sqrt{a} = \frac{ a + \sqrt{ab} } { \sqrt{a} + \sqrt{b} } = \frac{ X } { \sqrt{ X + Y } }$ .
Hence $a = \frac{ X^2 } { X+Y}$ .
Similarly, $b = \frac{ Y^2}{ X+Y }$ .
Thus, both $a$ and $b$ are rational.

Note: We didn't require $a \neq b$ .

As $a + \sqrt{ab} = \sqrt{a}(\sqrt{a} + \sqrt{b})$ and $b + \sqrt{ab} = \sqrt{b}(\sqrt{a} + \sqrt{b})$ , we see that $\sqrt{\dfrac{a}{b}} = x$ is rational and positive.

Hence, as $a \left (1+\dfrac{1}{x} \right)$ is rational, $a$ is rational. Similarly, $b$ is rational as $b(1+x)$ is.