Answer
$$\frac{1}{2+\sqrt{5}}=-2+\sqrt{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{2+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{2+\sqrt{5}}\frac{2-\sqrt{5}}{2-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2-\sqrt{5}}{4-2\sqrt{5}+2\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2-\sqrt{5}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-2+\sqrt{5}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-2+\sqrt{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 2- \sqrt{5}\right) = \color{blue}{1} \cdot2+\color{blue}{1} \cdot- \sqrt{5} = \\ = 2- \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 2 + \sqrt{5}\right) } \cdot \left( 2- \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot2+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 4- 2 \sqrt{5} + 2 \sqrt{5}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
| ⑤ | Remove 1 from denominator. |
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