Question

La expresión

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

es igual a:

The expression

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

equals:

2

2-\sqrt2

2+\sqrt2

2\sqrt2

\dfrac{\sqrt2}{2}

Racionaliza los dos términos recíprocos antes de sumarlos

Rationalize the two reciprocal terms before adding them

Los dos recíprocos se simplifican a cantidades cuya suma es

-\sqrt2

The two reciprocals simplify to quantities whose sum is

-\sqrt2

Al racionalizar se obtiene

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

Su suma es

-\sqrt2.

Por lo tanto, la expresión completa es

2+\sqrt2-\sqrt2=2.

Por lo tanto, la respuesta correcta es A.

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.