Answer
$$\frac{1-\sqrt{20}}{2+\sqrt{20}}=\frac{-11+3\sqrt{5}}{8}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1-\sqrt{20}}{2+\sqrt{20}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1-\sqrt{20}}{2+\sqrt{20}}\frac{2-\sqrt{20}}{2-\sqrt{20}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2-2\sqrt{5}-4\sqrt{5}+20}{4-4\sqrt{5}+4\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{22-6\sqrt{5}}{-16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{11-3\sqrt{5}}{-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-11+3\sqrt{5}}{8}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2- \sqrt{20}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 1- \sqrt{20}\right) } \cdot \left( 2- \sqrt{20}\right) = \color{blue}{1} \cdot2+\color{blue}{1} \cdot- \sqrt{20}\color{blue}{- \sqrt{20}} \cdot2\color{blue}{- \sqrt{20}} \cdot- \sqrt{20} = \\ = 2- 2 \sqrt{5}- 4 \sqrt{5} + 20 $$ Simplify denominator. $$ \color{blue}{ \left( 2 + \sqrt{20}\right) } \cdot \left( 2- \sqrt{20}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- \sqrt{20}+\color{blue}{ \sqrt{20}} \cdot2+\color{blue}{ \sqrt{20}} \cdot- \sqrt{20} = \\ = 4- 4 \sqrt{5} + 4 \sqrt{5}-20 $$ |
| 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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