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 If we write $\sqrt{2}+\frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{2}}$ in the form $\dfrac{a\sqrt{2} + b\sqrt{3}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$?

 May 9, 2022
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\(\quad\displaystyle \sqrt{2}+\frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{2}} \\= \sqrt 2 + \sqrt 3 + \dfrac 2{\sqrt 2} \\= 2 \sqrt 2 + \sqrt3\)

 

Now, the values of a, b, and c are obvious.

 May 9, 2022

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