Question
Answer
$$\dfrac{2}{2-5\sqrt{5}}=-\dfrac{4+10\sqrt{5}}{121}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{2-5\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{2-5\sqrt{5}}\frac{2+5\sqrt{5}}{2+5\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4+10\sqrt{5}}{4+10\sqrt{5}-10\sqrt{5}-125} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4+10\sqrt{5}}{-121} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{4+10\sqrt{5}}{121}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + 5 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( 2 + 5 \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot 5 \sqrt{5} = \\ = 4 + 10 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 2- 5 \sqrt{5}\right) } \cdot \left( 2 + 5 \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot 5 \sqrt{5}\color{blue}{- 5 \sqrt{5}} \cdot2\color{blue}{- 5 \sqrt{5}} \cdot 5 \sqrt{5} = \\ = 4 + 10 \sqrt{5}- 10 \sqrt{5}-125 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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