Answer
$$\frac{\sqrt{5}+\sqrt{11}}{2\sqrt{5}-\sqrt{11}}=\frac{7+\sqrt{55}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}+\sqrt{11}}{2\sqrt{5}-\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}+\sqrt{11}}{2\sqrt{5}-\sqrt{11}}\frac{2\sqrt{5}+\sqrt{11}}{2\sqrt{5}+\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10+\sqrt{55}+2\sqrt{55}+11}{20+2\sqrt{55}-2\sqrt{55}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{21+3\sqrt{55}}{9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{7+\sqrt{55}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5} + \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{5} + \sqrt{11}\right) } \cdot \left( 2 \sqrt{5} + \sqrt{11}\right) = \color{blue}{ \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot 2 \sqrt{5}+\color{blue}{ \sqrt{11}} \cdot \sqrt{11} = \\ = 10 + \sqrt{55} + 2 \sqrt{55} + 11 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{5}- \sqrt{11}\right) } \cdot \left( 2 \sqrt{5} + \sqrt{11}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot \sqrt{11}\color{blue}{- \sqrt{11}} \cdot 2 \sqrt{5}\color{blue}{- \sqrt{11}} \cdot \sqrt{11} = \\ = 20 + 2 \sqrt{55}- 2 \sqrt{55}-11 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
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