Making Rational Expressions

Question

Find all positive integers a and b such that

$$ \frac{\sqrt2 + \sqrt a}{\sqrt3 + \sqrt b}$$

is rational.

I tried Equating to some number r and squaring it, and the answer is a= 3 b =2, but im not sure if those are all

Answer 1

Consider that we require that $$ \frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}} = \frac{m}n $$ for some positive integers $m$ and $n$. Now, we can write this as $$ n\sqrt{2}+n\sqrt{a} = m\sqrt{3}+m\sqrt{b} $$ From here, we can easily recognise that $a$ must be of the form $3\alpha^2$ and $b$ must be of the form $2\beta^2$, as both $\sqrt{2}$ and $\sqrt{3}$ must be present on both sides. Making this substitution, we have $$ (n-\beta m)\sqrt{2} = (m-\alpha n)\sqrt{3} $$ As both bracketed terms are integers, we conclude that they must both be zero. Therefore, we have $$ n-\beta m = 0\\ m-\alpha n = 0 $$ So we have $n=\beta m$, which gives us $m=\alpha\beta m$, or $\alpha\beta = 1$. But we also have that $\alpha$ and $\beta$ are positive integers - therefore, $\alpha=\beta=1$, which gives $m=n$.

Therefore, the only solution is $a=3$, $b=2$.