1970 AHSME Problems/Problem 18

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Problem

$\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}$ is equal to

$\text{(A) } 2\quad \text{(B) } 2\sqrt{3}\quad \text{(C) } 4\sqrt{2}\quad \text{(D) } \sqrt{6}\quad \text{(E) } 2\sqrt{2}$

Solution

Square the expression:

$(\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}})^2=3+\sqrt{2}-2\sqrt{(3+\sqrt{2})(3-\sqrt{2})}+3-2\sqrt{2}=6-2\sqrt{9-8}=6-2\sqrt{1}=4$

$\Rightarrow\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}=\sqrt{4}=2\Rightarrow\boxed{A}$

See also

1970 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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