Answer
$$\frac{4+\sqrt{10}}{5-\sqrt{10}}=\frac{10+3\sqrt{10}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4+\sqrt{10}}{5-\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4+\sqrt{10}}{5-\sqrt{10}}\frac{5+\sqrt{10}}{5+\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20+4\sqrt{10}+5\sqrt{10}+10}{25+5\sqrt{10}-5\sqrt{10}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{30+9\sqrt{10}}{15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{10+3\sqrt{10}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4 + \sqrt{10}\right) } \cdot \left( 5 + \sqrt{10}\right) = \color{blue}{4} \cdot5+\color{blue}{4} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot5+\color{blue}{ \sqrt{10}} \cdot \sqrt{10} = \\ = 20 + 4 \sqrt{10} + 5 \sqrt{10} + 10 $$ Simplify denominator. $$ \color{blue}{ \left( 5- \sqrt{10}\right) } \cdot \left( 5 + \sqrt{10}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot \sqrt{10}\color{blue}{- \sqrt{10}} \cdot5\color{blue}{- \sqrt{10}} \cdot \sqrt{10} = \\ = 25 + 5 \sqrt{10}- 5 \sqrt{10}-10 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
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