# 2004 Pan African MO Problems/Problem 2

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$\sqrt{4-2\sqrt{3}} = a\sqrt{3}-b$. Through guess and check with small numbers, $a = 1$ and $b = 1$. So $\sqrt{4-2\sqrt{3}} = \sqrt{3}-1$.

$\sqrt{97-56\sqrt{3}} = a-b\sqrt{3}$. Through prime factorization, $a = 7$ and $b = -4$. So $\sqrt{97-56\sqrt{3}} = 7-4\sqrt{3}$.

Value of $4\sqrt{4-2\sqrt{3}} + \sqrt{97-56\sqrt{3}} = (4\sqrt{3}-4) + (7-4\sqrt{3}) = 3$