Answer
$$\frac{2\sqrt{5}}{4\sqrt{5}+9}=-40+18\sqrt{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{5}}{4\sqrt{5}+9}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{5}}{4\sqrt{5}+9}\frac{4\sqrt{5}-9}{4\sqrt{5}-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{40-18\sqrt{5}}{80-36\sqrt{5}+36\sqrt{5}-81} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{40-18\sqrt{5}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-40+18\sqrt{5}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-40+18\sqrt{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{5}-9} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 \sqrt{5} } \cdot \left( 4 \sqrt{5}-9\right) = \color{blue}{ 2 \sqrt{5}} \cdot 4 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot-9 = \\ = 40- 18 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{5} + 9\right) } \cdot \left( 4 \sqrt{5}-9\right) = \color{blue}{ 4 \sqrt{5}} \cdot 4 \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot-9+\color{blue}{9} \cdot 4 \sqrt{5}+\color{blue}{9} \cdot-9 = \\ = 80- 36 \sqrt{5} + 36 \sqrt{5}-81 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
| ⑤ | Remove 1 from denominator. |
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