2004 Pan African MO Problems/Problem 2

Problem

Is $4\sqrt{4-2\sqrt{3}}+\sqrt{97-56\sqrt{3}}$ an integer?

$\sqrt{4-2\sqrt{3}} = a\sqrt{3}-b$ . 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 $4\sqrt{4-2\sqrt{3}} + \sqrt{97-56\sqrt{3}} = (4\sqrt{3}-4) + (7-4\sqrt{3}) = 3$ .

2004 Pan African MO (Problems)
Preceded by Problem 1	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 3
All Pan African MO Problems and Solutions