Answer
$$\frac{3-\sqrt{5}}{3+2\sqrt{5}}=\frac{-19+9\sqrt{5}}{11}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3-\sqrt{5}}{3+2\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3-\sqrt{5}}{3+2\sqrt{5}}\frac{3-2\sqrt{5}}{3-2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9-6\sqrt{5}-3\sqrt{5}+10}{9-6\sqrt{5}+6\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{19-9\sqrt{5}}{-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-19+9\sqrt{5}}{11}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- 2 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3- \sqrt{5}\right) } \cdot \left( 3- 2 \sqrt{5}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- 2 \sqrt{5}\color{blue}{- \sqrt{5}} \cdot3\color{blue}{- \sqrt{5}} \cdot- 2 \sqrt{5} = \\ = 9- 6 \sqrt{5}- 3 \sqrt{5} + 10 $$ Simplify denominator. $$ \color{blue}{ \left( 3 + 2 \sqrt{5}\right) } \cdot \left( 3- 2 \sqrt{5}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot3+\color{blue}{ 2 \sqrt{5}} \cdot- 2 \sqrt{5} = \\ = 9- 6 \sqrt{5} + 6 \sqrt{5}-20 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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