The expression 2+√2+12+√2+1√2-2 equals:

1958 AMC 12 Problem 5

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The expression

2+\sqrt2+\dfrac1{2+\sqrt2}+\dfrac1{\sqrt2-2}

equals:

2

2-\sqrt2

2+\sqrt2

2\sqrt2

\dfrac{\sqrt2}{2}

Rationalize the two reciprocal terms before adding them

The two reciprocals simplify to quantities whose sum is

-\sqrt2

Rationalizing gives

\begin{aligned} \frac1{2+\sqrt2}&=\frac{2-\sqrt2}{2},\\ \frac1{\sqrt2-2}&=-\frac{2+\sqrt2}{2}. \end{aligned}

Their sum is

-\sqrt2.

Therefore the whole expression is

2+\sqrt2-\sqrt2=2.

Thus, the correct answer is A.