Answer
$$\frac{\frac{3-2\sqrt{5}}{1+3\sqrt{5}}}{\frac{3-2\sqrt{5}}{1+3\sqrt{5}}}=1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\frac{3-2\sqrt{5}}{1+3\sqrt{5}}}{\frac{3-2\sqrt{5}}{1+3\sqrt{5}}}& \xlongequal{ }\frac{3-2\sqrt{5}}{1+3\sqrt{5}}\frac{1+3\sqrt{5}}{3-2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3+9\sqrt{5}-2\sqrt{5}-30}{3-2\sqrt{5}+9\sqrt{5}-30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-27+7\sqrt{5}}{-27+7\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-27+7\sqrt{5}}{-27+7\sqrt{5}}\frac{-27-7\sqrt{5}}{-27-7\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{729+189\sqrt{5}-189\sqrt{5}-245}{729+189\sqrt{5}-189\sqrt{5}-245} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{484}{484} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}} \frac{ 484 : \color{orangered}{ 484 } }{ 484 : \color{orangered}{ 484 }} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{1}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}1\end{aligned} $$ | |
| ① | $$ \color{blue}{ \left( 3- 2 \sqrt{5}\right) } \cdot \left( 1 + 3 \sqrt{5}\right) = \color{blue}{3} \cdot1+\color{blue}{3} \cdot 3 \sqrt{5}\color{blue}{- 2 \sqrt{5}} \cdot1\color{blue}{- 2 \sqrt{5}} \cdot 3 \sqrt{5} = \\ = 3 + 9 \sqrt{5}- 2 \sqrt{5}-30 $$$$ \color{blue}{ \left( 1 + 3 \sqrt{5}\right) } \cdot \left( 3- 2 \sqrt{5}\right) = \color{blue}{1} \cdot3+\color{blue}{1} \cdot- 2 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot3+\color{blue}{ 3 \sqrt{5}} \cdot- 2 \sqrt{5} = \\ = 3- 2 \sqrt{5} + 9 \sqrt{5}-30 $$ |
| ② | Simplify numerator and denominator |
| ③ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -27- 7 \sqrt{5}} $$. |
| ④ | Multiply in a numerator. $$ \color{blue}{ \left( -27 + 7 \sqrt{5}\right) } \cdot \left( -27- 7 \sqrt{5}\right) = \color{blue}{-27} \cdot-27\color{blue}{-27} \cdot- 7 \sqrt{5}+\color{blue}{ 7 \sqrt{5}} \cdot-27+\color{blue}{ 7 \sqrt{5}} \cdot- 7 \sqrt{5} = \\ = 729 + 189 \sqrt{5}- 189 \sqrt{5}-245 $$ Simplify denominator. $$ \color{blue}{ \left( -27 + 7 \sqrt{5}\right) } \cdot \left( -27- 7 \sqrt{5}\right) = \color{blue}{-27} \cdot-27\color{blue}{-27} \cdot- 7 \sqrt{5}+\color{blue}{ 7 \sqrt{5}} \cdot-27+\color{blue}{ 7 \sqrt{5}} \cdot- 7 \sqrt{5} = \\ = 729 + 189 \sqrt{5}- 189 \sqrt{5}-245 $$ |
| ⑤ | Simplify numerator and denominator |
| ⑥ | Divide both the top and bottom numbers by $ \color{orangered}{ 484 } $. |
| ⑦ | Remove 1 from denominator. |
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