Answer
$$\frac{2+6\sqrt{5}}{2-6\sqrt{5}}=-\frac{23+3\sqrt{5}}{22}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2+6\sqrt{5}}{2-6\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2+6\sqrt{5}}{2-6\sqrt{5}}\frac{2+6\sqrt{5}}{2+6\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4+12\sqrt{5}+12\sqrt{5}+180}{4+12\sqrt{5}-12\sqrt{5}-180} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{184+24\sqrt{5}}{-176} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{23+3\sqrt{5}}{-22} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{23+3\sqrt{5}}{22}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + 6 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 + 6 \sqrt{5}\right) } \cdot \left( 2 + 6 \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot 6 \sqrt{5}+\color{blue}{ 6 \sqrt{5}} \cdot2+\color{blue}{ 6 \sqrt{5}} \cdot 6 \sqrt{5} = \\ = 4 + 12 \sqrt{5} + 12 \sqrt{5} + 180 $$ Simplify denominator. $$ \color{blue}{ \left( 2- 6 \sqrt{5}\right) } \cdot \left( 2 + 6 \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot 6 \sqrt{5}\color{blue}{- 6 \sqrt{5}} \cdot2\color{blue}{- 6 \sqrt{5}} \cdot 6 \sqrt{5} = \\ = 4 + 12 \sqrt{5}- 12 \sqrt{5}-180 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 8. |
| ⑤ | Place a negative sign in front of a fraction. |
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