Answer
$$\frac{5\sqrt{5}+\sqrt{10}}{10\sqrt{5}-\sqrt{10}}=\frac{52+15\sqrt{2}}{98}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{5}+\sqrt{10}}{10\sqrt{5}-\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{5}+\sqrt{10}}{10\sqrt{5}-\sqrt{10}}\frac{10\sqrt{5}+\sqrt{10}}{10\sqrt{5}+\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{250+25\sqrt{2}+50\sqrt{2}+10}{500+50\sqrt{2}-50\sqrt{2}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{260+75\sqrt{2}}{490} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{52+15\sqrt{2}}{98}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 10 \sqrt{5} + \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 \sqrt{5} + \sqrt{10}\right) } \cdot \left( 10 \sqrt{5} + \sqrt{10}\right) = \color{blue}{ 5 \sqrt{5}} \cdot 10 \sqrt{5}+\color{blue}{ 5 \sqrt{5}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot 10 \sqrt{5}+\color{blue}{ \sqrt{10}} \cdot \sqrt{10} = \\ = 250 + 25 \sqrt{2} + 50 \sqrt{2} + 10 $$ Simplify denominator. $$ \color{blue}{ \left( 10 \sqrt{5}- \sqrt{10}\right) } \cdot \left( 10 \sqrt{5} + \sqrt{10}\right) = \color{blue}{ 10 \sqrt{5}} \cdot 10 \sqrt{5}+\color{blue}{ 10 \sqrt{5}} \cdot \sqrt{10}\color{blue}{- \sqrt{10}} \cdot 10 \sqrt{5}\color{blue}{- \sqrt{10}} \cdot \sqrt{10} = \\ = 500 + 50 \sqrt{2}- 50 \sqrt{2}-10 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
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