Answer
$$\frac{1}{(1-\sqrt{5})^2}=\frac{3+\sqrt{5}}{8}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{(1-\sqrt{5})^2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{1-\sqrt{5}-\sqrt{5}+5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1}{6-2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1}{6-2\sqrt{5}}\frac{6+2\sqrt{5}}{6+2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6+2\sqrt{5}}{36+12\sqrt{5}-12\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{6+2\sqrt{5}}{16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{3+\sqrt{5}}{8}\end{aligned} $$ | |
| ① | $$ (1-\sqrt{5})^2 = \left( 1- \sqrt{5} \right) \cdot \left( 1- \sqrt{5} \right) = 1- \sqrt{5}- \sqrt{5} + 5 $$ |
| ② | Simplify numerator and denominator |
| ③ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6 + 2 \sqrt{5}} $$. |
| ④ | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 6 + 2 \sqrt{5}\right) = \color{blue}{1} \cdot6+\color{blue}{1} \cdot 2 \sqrt{5} = \\ = 6 + 2 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 6- 2 \sqrt{5}\right) } \cdot \left( 6 + 2 \sqrt{5}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot 2 \sqrt{5}\color{blue}{- 2 \sqrt{5}} \cdot6\color{blue}{- 2 \sqrt{5}} \cdot 2 \sqrt{5} = \\ = 36 + 12 \sqrt{5}- 12 \sqrt{5}-20 $$ |
| ⑤ | Simplify numerator and denominator |
| ⑥ | Divide both numerator and denominator by 2. |
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