Question
Answer
$$\dfrac{\sqrt{5}}{2-2\sqrt{5}}=-\dfrac{\sqrt{5}+5}{8}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}}{2-2\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}}{2-2\sqrt{5}}\frac{2+2\sqrt{5}}{2+2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{5}+10}{4+4\sqrt{5}-4\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{5}+10}{-16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{5}+5}{-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{\sqrt{5}+5}{8}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + 2 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{5} } \cdot \left( 2 + 2 \sqrt{5}\right) = \color{blue}{ \sqrt{5}} \cdot2+\color{blue}{ \sqrt{5}} \cdot 2 \sqrt{5} = \\ = 2 \sqrt{5} + 10 $$ Simplify denominator. $$ \color{blue}{ \left( 2- 2 \sqrt{5}\right) } \cdot \left( 2 + 2 \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot 2 \sqrt{5}\color{blue}{- 2 \sqrt{5}} \cdot2\color{blue}{- 2 \sqrt{5}} \cdot 2 \sqrt{5} = \\ = 4 + 4 \sqrt{5}- 4 \sqrt{5}-20 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Place a negative sign in front of a fraction. |
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