Answer
$$\frac{3\sqrt{11}}{3\sqrt{11}+10}=-99+30\sqrt{11}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{11}}{3\sqrt{11}+10}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{11}}{3\sqrt{11}+10}\frac{3\sqrt{11}-10}{3\sqrt{11}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{99-30\sqrt{11}}{99-30\sqrt{11}+30\sqrt{11}-100} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{99-30\sqrt{11}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-99+30\sqrt{11}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-99+30\sqrt{11}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{11}-10} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 \sqrt{11} } \cdot \left( 3 \sqrt{11}-10\right) = \color{blue}{ 3 \sqrt{11}} \cdot 3 \sqrt{11}+\color{blue}{ 3 \sqrt{11}} \cdot-10 = \\ = 99- 30 \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{11} + 10\right) } \cdot \left( 3 \sqrt{11}-10\right) = \color{blue}{ 3 \sqrt{11}} \cdot 3 \sqrt{11}+\color{blue}{ 3 \sqrt{11}} \cdot-10+\color{blue}{10} \cdot 3 \sqrt{11}+\color{blue}{10} \cdot-10 = \\ = 99- 30 \sqrt{11} + 30 \sqrt{11}-100 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
| ⑤ | Remove 1 from denominator. |
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