Answer
$$\frac{\sqrt{11}+3}{\sqrt{11}-3}=10+3\sqrt{11}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{11}+3}{\sqrt{11}-3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{11}+3}{\sqrt{11}-3}\frac{\sqrt{11}+3}{\sqrt{11}+3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{11+3\sqrt{11}+3\sqrt{11}+9}{11+3\sqrt{11}-3\sqrt{11}-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20+6\sqrt{11}}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{10+3\sqrt{11}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}10+3\sqrt{11}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{11} + 3} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{11} + 3\right) } \cdot \left( \sqrt{11} + 3\right) = \color{blue}{ \sqrt{11}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot3+\color{blue}{3} \cdot \sqrt{11}+\color{blue}{3} \cdot3 = \\ = 11 + 3 \sqrt{11} + 3 \sqrt{11} + 9 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{11}-3\right) } \cdot \left( \sqrt{11} + 3\right) = \color{blue}{ \sqrt{11}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot3\color{blue}{-3} \cdot \sqrt{11}\color{blue}{-3} \cdot3 = \\ = 11 + 3 \sqrt{11}- 3 \sqrt{11}-9 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Remove 1 from denominator. |
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