Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer

Simplify: 21.5+2−(1.5+2)2 \sqrt{1.5+\sqrt{2}}-(1.5+\sqrt{2}).

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The given expression equals 6+42−(1.5+2)=6+28−(1.5+2)\sqrt{6+4 \sqrt{2}}-(1.5+\sqrt{2})=\sqrt{6+2 \sqrt{8}}-(1.5+\sqrt{2}). But on inspection, we see that (2+4)2=6+28(\sqrt{2}+\sqrt{4})^{2}=6+2 \sqrt{8}, so the answer is (2+4)−(1.5+2)=2−3/2=1/2(\sqrt{2}+\sqrt{4})-(1.5+\sqrt{2})=2-3 / 2=1 / 2.

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