Answer
$$\frac{2}{10-\sqrt{2}}=\frac{10+\sqrt{2}}{49}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{10-\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{10-\sqrt{2}}\frac{10+\sqrt{2}}{10+\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20+2\sqrt{2}}{100+10\sqrt{2}-10\sqrt{2}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20+2\sqrt{2}}{98} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{10+\sqrt{2}}{49}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 10 + \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( 10 + \sqrt{2}\right) = \color{blue}{2} \cdot10+\color{blue}{2} \cdot \sqrt{2} = \\ = 20 + 2 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 10- \sqrt{2}\right) } \cdot \left( 10 + \sqrt{2}\right) = \color{blue}{10} \cdot10+\color{blue}{10} \cdot \sqrt{2}\color{blue}{- \sqrt{2}} \cdot10\color{blue}{- \sqrt{2}} \cdot \sqrt{2} = \\ = 100 + 10 \sqrt{2}- 10 \sqrt{2}-2 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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