Answer
$$\frac{\sqrt{11}+1}{\sqrt{11}-1}=\frac{6+\sqrt{11}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{11}+1}{\sqrt{11}-1}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{11}+1}{\sqrt{11}-1}\frac{\sqrt{11}+1}{\sqrt{11}+1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{11+\sqrt{11}+\sqrt{11}+1}{11+\sqrt{11}-\sqrt{11}-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12+2\sqrt{11}}{10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6+\sqrt{11}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{11} + 1} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{11} + 1\right) } \cdot \left( \sqrt{11} + 1\right) = \color{blue}{ \sqrt{11}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot1+\color{blue}{1} \cdot \sqrt{11}+\color{blue}{1} \cdot1 = \\ = 11 + \sqrt{11} + \sqrt{11} + 1 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{11}-1\right) } \cdot \left( \sqrt{11} + 1\right) = \color{blue}{ \sqrt{11}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot1\color{blue}{-1} \cdot \sqrt{11}\color{blue}{-1} \cdot1 = \\ = 11 + \sqrt{11}- \sqrt{11}-1 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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