Question:

Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. Multiply by $\dfrac{4+2\sqrt{3}}{4+2\sqrt{3}}$: denominator becomes $16-12=4$. Numerator = $(\sqrt{3}+4\sqrt{2})(4+2\sqrt{3}) = 4\sqrt{3}+2\cdot3+16\sqrt{2}+8\sqrt{6} = 6+4\sqrt{3}+16\sqrt{2}+8\sqrt{6}$. Dividing by 4 and reading off $a,b,c$ gives $a+b+c=70$. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.