Answer
$$\frac{\sqrt{5}}{8+10\sqrt{2}}=\frac{-4\sqrt{5}+5\sqrt{10}}{68}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}}{8+10\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}}{8+10\sqrt{2}}\frac{8-10\sqrt{2}}{8-10\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8\sqrt{5}-10\sqrt{10}}{64-80\sqrt{2}+80\sqrt{2}-200} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8\sqrt{5}-10\sqrt{10}}{-136} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4\sqrt{5}-5\sqrt{10}}{-68} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-4\sqrt{5}+5\sqrt{10}}{68}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 8- 10 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{5} } \cdot \left( 8- 10 \sqrt{2}\right) = \color{blue}{ \sqrt{5}} \cdot8+\color{blue}{ \sqrt{5}} \cdot- 10 \sqrt{2} = \\ = 8 \sqrt{5}- 10 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 8 + 10 \sqrt{2}\right) } \cdot \left( 8- 10 \sqrt{2}\right) = \color{blue}{8} \cdot8+\color{blue}{8} \cdot- 10 \sqrt{2}+\color{blue}{ 10 \sqrt{2}} \cdot8+\color{blue}{ 10 \sqrt{2}} \cdot- 10 \sqrt{2} = \\ = 64- 80 \sqrt{2} + 80 \sqrt{2}-200 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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