Answer
$$\frac{1}{13+\sqrt{2}}=\frac{13-\sqrt{2}}{167}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{13+\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{13+\sqrt{2}}\frac{13-\sqrt{2}}{13-\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{13-\sqrt{2}}{169-13\sqrt{2}+13\sqrt{2}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{13-\sqrt{2}}{167}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 13- \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 13- \sqrt{2}\right) = \color{blue}{1} \cdot13+\color{blue}{1} \cdot- \sqrt{2} = \\ = 13- \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 13 + \sqrt{2}\right) } \cdot \left( 13- \sqrt{2}\right) = \color{blue}{13} \cdot13+\color{blue}{13} \cdot- \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot13+\color{blue}{ \sqrt{2}} \cdot- \sqrt{2} = \\ = 169- 13 \sqrt{2} + 13 \sqrt{2}-2 $$ |
| ③ | Simplify numerator and denominator |
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