Equations with Powers, Roots, and Radicals

It turns out $\sqrt{2 + \sqrt{3}} = \frac{a + \sqrt{b}}{\sqrt{c}}$, where $a,b,c$ are positive integers with no common factor, and $b,c$ are "square-free" (meaning they are not divisible by any of the squares $2^2,3^2,4^2,\dots$ greater than $1$). What is $a+b+c$?

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(Hint: Square both sides and simplify.)

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Source: Math Prize for Girls 2011