Answer
$$\frac{5-\sqrt{10}}{3+\sqrt{10}}=-25+8\sqrt{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5-\sqrt{10}}{3+\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5-\sqrt{10}}{3+\sqrt{10}}\frac{3-\sqrt{10}}{3-\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{15-5\sqrt{10}-3\sqrt{10}+10}{9-3\sqrt{10}+3\sqrt{10}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{25-8\sqrt{10}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-25+8\sqrt{10}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-25+8\sqrt{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5- \sqrt{10}\right) } \cdot \left( 3- \sqrt{10}\right) = \color{blue}{5} \cdot3+\color{blue}{5} \cdot- \sqrt{10}\color{blue}{- \sqrt{10}} \cdot3\color{blue}{- \sqrt{10}} \cdot- \sqrt{10} = \\ = 15- 5 \sqrt{10}- 3 \sqrt{10} + 10 $$ Simplify denominator. $$ \color{blue}{ \left( 3 + \sqrt{10}\right) } \cdot \left( 3- \sqrt{10}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot3+\color{blue}{ \sqrt{10}} \cdot- \sqrt{10} = \\ = 9- 3 \sqrt{10} + 3 \sqrt{10}-10 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
| ⑤ | Remove 1 from denominator. |
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