I found the following problem on a Olympiad question paper:
For which $a,b\in \mathbb{N},$ is $$\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$$ a rational number.
I am unable to solve it.
Any help will be appreciated.
2026-03-27 00:05:06.1774569906
For which $a,b\in \mathbb{N},$ is $\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is a rational number.
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Well, $$\begin{align}\dfrac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}} ~=~& \dfrac{(\sqrt{2}+\sqrt{a})(\sqrt{3}-\sqrt{b})}{(\sqrt{3}+\sqrt{b})(\sqrt{3}-\sqrt{b})} & \textsf{except when }b=3 \\[2ex] =~&\dfrac{\sqrt{6}+\sqrt{3a}-\sqrt{2b}-\sqrt{ab}}{3-b} \end{align}$$
Then, when might this be rational? ...