Malone Flynn
07/12/2024 · Elementary School
Relacionar el siguiente denominadores: \( \frac{1}{2+\sqrt{2}} \)
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Tutor-Verified Answer
Step-by-step Solution
Para racionalizar el denominador de la fracción \( \frac{1}{2+\sqrt{2}} \), multiplicamos tanto el numerador como el denominador por el conjugado del denominador, que en este caso es \( 2 - \sqrt{2} \).
Así que hacemos lo siguiente:
\[
\frac{1}{2+\sqrt{2}} \cdot \frac{2-\sqrt{2}}{2-\sqrt{2}} = \frac{2 - \sqrt{2}}{(2+\sqrt{2})(2-\sqrt{2})}
\]
Ahora, calculamos el denominador:
\[
(2+\sqrt{2})(2-\sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2
\]
Por lo tanto, la fracción se convierte en:
\[
\frac{2 - \sqrt{2}}{2}
\]
Podemos simplificar esto dividiendo cada término en el numerador por 2:
\[
\frac{2}{2} - \frac{\sqrt{2}}{2} = 1 - \frac{\sqrt{2}}{2}
\]
Así que la forma racionalizada de \( \frac{1}{2+\sqrt{2}} \) es:
\[
1 - \frac{\sqrt{2}}{2}
\]
Quick Answer
La forma racionalizada de \( \frac{1}{2+\sqrt{2}} \) es \( 1 - \frac{\sqrt{2}}{2} \).
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