Question
Answer
$$\dfrac{2+4\sqrt{5}}{\sqrt{5}+5}=\dfrac{9\sqrt{5}-5}{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2+4\sqrt{5}}{\sqrt{5}+5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2+4\sqrt{5}}{\sqrt{5}+5}\frac{\sqrt{5}-5}{\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{5}-10+20-20\sqrt{5}}{5-5\sqrt{5}+5\sqrt{5}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-18\sqrt{5}+10}{-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-9\sqrt{5}+5}{-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{9\sqrt{5}-5}{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5}-5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 + 4 \sqrt{5}\right) } \cdot \left( \sqrt{5}-5\right) = \color{blue}{2} \cdot \sqrt{5}+\color{blue}{2} \cdot-5+\color{blue}{ 4 \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot-5 = \\ = 2 \sqrt{5}-10 + 20- 20 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5} + 5\right) } \cdot \left( \sqrt{5}-5\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot-5+\color{blue}{5} \cdot \sqrt{5}+\color{blue}{5} \cdot-5 = \\ = 5- 5 \sqrt{5} + 5 \sqrt{5}-25 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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