Answer
$$\frac{1}{5-4\sqrt{5}+4}=9+4\sqrt{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{5-4\sqrt{5}+4}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{9-4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1}{9-4\sqrt{5}}\frac{9+4\sqrt{5}}{9+4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9+4\sqrt{5}}{81+36\sqrt{5}-36\sqrt{5}-80} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{9+4\sqrt{5}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}9+4\sqrt{5}\end{aligned} $$ | |
| ① | Simplify numerator and denominator |
| ② | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 9 + 4 \sqrt{5}} $$. |
| ③ | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 9 + 4 \sqrt{5}\right) = \color{blue}{1} \cdot9+\color{blue}{1} \cdot 4 \sqrt{5} = \\ = 9 + 4 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 9- 4 \sqrt{5}\right) } \cdot \left( 9 + 4 \sqrt{5}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot 4 \sqrt{5}\color{blue}{- 4 \sqrt{5}} \cdot9\color{blue}{- 4 \sqrt{5}} \cdot 4 \sqrt{5} = \\ = 81 + 36 \sqrt{5}- 36 \sqrt{5}-80 $$ |
| ④ | Simplify numerator and denominator |
| ⑤ | Remove 1 from denominator. |
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