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 $a+b+c$?
I notice that the given expression has four terms, two of which are irrational and two of which are rational. I wonder if I can simplify the expression by combining the rational terms somehow. I see that the rational terms have square roots in their denominators, so I can try to rationalize them by multiplying them by their conjugates. For example, $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ and $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$. So I can rewrite the expression as $\sqrt{2} + \frac{\sqrt{2}}{2} + \sqrt{3} + \frac{\sqrt{3}}{3}$. Now I can combine the terms with the same radicals by adding their coefficients. This gives me $\frac{3\sqrt{2}}{2} + \frac{4\sqrt{3}}{3}$. I notice that the denominators are different, so I can try to find a common denominator to make the expression simpler. The least common multiple of 2 and 3 is 6, so I can multiply the first term by 3 and the second term by 2 to get the same denominator. This gives me $\frac{9\sqrt{2} + 8\sqrt{3}}{6}$. Now I have the expression in the desired form, where $a = 9$, $b = 8$, and $c = 6$. I can add these values to get the final answer. 