Answer
$$\frac{2\sqrt{5}+5}{1+2\sqrt{5}}=\frac{8\sqrt{5}+15}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{5}+5}{1+2\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{5}+5}{1+2\sqrt{5}}\frac{1-2\sqrt{5}}{1-2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{5}-20+5-10\sqrt{5}}{1-2\sqrt{5}+2\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-8\sqrt{5}-15}{-19} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{8\sqrt{5}+15}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 1- 2 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 \sqrt{5} + 5\right) } \cdot \left( 1- 2 \sqrt{5}\right) = \color{blue}{ 2 \sqrt{5}} \cdot1+\color{blue}{ 2 \sqrt{5}} \cdot- 2 \sqrt{5}+\color{blue}{5} \cdot1+\color{blue}{5} \cdot- 2 \sqrt{5} = \\ = 2 \sqrt{5}-20 + 5- 10 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 1 + 2 \sqrt{5}\right) } \cdot \left( 1- 2 \sqrt{5}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot- 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot1+\color{blue}{ 2 \sqrt{5}} \cdot- 2 \sqrt{5} = \\ = 1- 2 \sqrt{5} + 2 \sqrt{5}-20 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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