Answer
$$\frac{\sqrt{5}+6}{\sqrt{5}-6}=-\frac{41+12\sqrt{5}}{31}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}+6}{\sqrt{5}-6}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}+6}{\sqrt{5}-6}\frac{\sqrt{5}+6}{\sqrt{5}+6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5+6\sqrt{5}+6\sqrt{5}+36}{5+6\sqrt{5}-6\sqrt{5}-36} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{41+12\sqrt{5}}{-31} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{41+12\sqrt{5}}{31}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5} + 6} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{5} + 6\right) } \cdot \left( \sqrt{5} + 6\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot6+\color{blue}{6} \cdot \sqrt{5}+\color{blue}{6} \cdot6 = \\ = 5 + 6 \sqrt{5} + 6 \sqrt{5} + 36 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5}-6\right) } \cdot \left( \sqrt{5} + 6\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot6\color{blue}{-6} \cdot \sqrt{5}\color{blue}{-6} \cdot6 = \\ = 5 + 6 \sqrt{5}- 6 \sqrt{5}-36 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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