Answer
$$\frac{11}{3+\sqrt{11}}=\frac{-33+11\sqrt{11}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{11}{3+\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{11}{3+\sqrt{11}}\frac{3-\sqrt{11}}{3-\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{33-11\sqrt{11}}{9-3\sqrt{11}+3\sqrt{11}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{33-11\sqrt{11}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-33+11\sqrt{11}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 11 } \cdot \left( 3- \sqrt{11}\right) = \color{blue}{11} \cdot3+\color{blue}{11} \cdot- \sqrt{11} = \\ = 33- 11 \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ \left( 3 + \sqrt{11}\right) } \cdot \left( 3- \sqrt{11}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot3+\color{blue}{ \sqrt{11}} \cdot- \sqrt{11} = \\ = 9- 3 \sqrt{11} + 3 \sqrt{11}-11 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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