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