If we write $\sqrt{2}+\frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{3}}$ 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 $a+b+c$?

We are tasked with simplifying the expression:

\[
\sqrt{2} + \frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{3}}
\]

and rewriting it in the form $ \dfrac{a\sqrt{2} + b\sqrt{3}}{c} $, where $a$, $b$, and $c$ are positive integers and $c$ is as small as

If we write $\sqrt{2} + \frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{3}}$ 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 $a + b + c$?

Discover how to simplify the expression $\sqrt{2} + \frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{3}}$ into the form $\dfrac{a\sqrt{2} + b\sqrt{3}}{c}$, where $a$, $b$, and $c$ are positive integers. This problem involves key algebraic concepts like simplifying radicals and basic algebraic operations, suitable for Grades 9-12.

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