Answer
$$\frac{2-\sqrt{5}}{2+3\sqrt{5}}=\frac{-19+8\sqrt{5}}{41}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2-\sqrt{5}}{2+3\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2-\sqrt{5}}{2+3\sqrt{5}}\frac{2-3\sqrt{5}}{2-3\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4-6\sqrt{5}-2\sqrt{5}+15}{4-6\sqrt{5}+6\sqrt{5}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{19-8\sqrt{5}}{-41} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-19+8\sqrt{5}}{41}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2- 3 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2- \sqrt{5}\right) } \cdot \left( 2- 3 \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- 3 \sqrt{5}\color{blue}{- \sqrt{5}} \cdot2\color{blue}{- \sqrt{5}} \cdot- 3 \sqrt{5} = \\ = 4- 6 \sqrt{5}- 2 \sqrt{5} + 15 $$ Simplify denominator. $$ \color{blue}{ \left( 2 + 3 \sqrt{5}\right) } \cdot \left( 2- 3 \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot2+\color{blue}{ 3 \sqrt{5}} \cdot- 3 \sqrt{5} = \\ = 4- 6 \sqrt{5} + 6 \sqrt{5}-45 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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