Answer
$$\frac{\sqrt{11}-\sqrt{5}}{\sqrt{11}+\sqrt{5}}=\frac{8-\sqrt{55}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{11}-\sqrt{5}}{\sqrt{11}+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{11}-\sqrt{5}}{\sqrt{11}+\sqrt{5}}\frac{\sqrt{11}-\sqrt{5}}{\sqrt{11}-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{11-\sqrt{55}-\sqrt{55}+5}{11-\sqrt{55}+\sqrt{55}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{16-2\sqrt{55}}{6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{8-\sqrt{55}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{11}- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{11}- \sqrt{5}\right) } \cdot \left( \sqrt{11}- \sqrt{5}\right) = \color{blue}{ \sqrt{11}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot- \sqrt{5}\color{blue}{- \sqrt{5}} \cdot \sqrt{11}\color{blue}{- \sqrt{5}} \cdot- \sqrt{5} = \\ = 11- \sqrt{55}- \sqrt{55} + 5 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{11} + \sqrt{5}\right) } \cdot \left( \sqrt{11}- \sqrt{5}\right) = \color{blue}{ \sqrt{11}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot \sqrt{11}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 11- \sqrt{55} + \sqrt{55}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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