Answer
$$\frac{\sqrt{2}}{2-\sqrt{2}}=\sqrt{2}+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{2-\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{2-\sqrt{2}}\frac{2+\sqrt{2}}{2+\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{2}+2}{4+2\sqrt{2}-2\sqrt{2}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{2}+2}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{2}+1}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\sqrt{2}+1\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( 2 + \sqrt{2}\right) = \color{blue}{ \sqrt{2}} \cdot2+\color{blue}{ \sqrt{2}} \cdot \sqrt{2} = \\ = 2 \sqrt{2} + 2 $$ Simplify denominator. $$ \color{blue}{ \left( 2- \sqrt{2}\right) } \cdot \left( 2 + \sqrt{2}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot \sqrt{2}\color{blue}{- \sqrt{2}} \cdot2\color{blue}{- \sqrt{2}} \cdot \sqrt{2} = \\ = 4 + 2 \sqrt{2}- 2 \sqrt{2}-2 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Remove 1 from denominator. |
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