Answer
$$\frac{11}{3+\sqrt{17}}=\frac{-33+11\sqrt{17}}{8}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{11}{3+\sqrt{17}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{11}{3+\sqrt{17}}\frac{3-\sqrt{17}}{3-\sqrt{17}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{33-11\sqrt{17}}{9-3\sqrt{17}+3\sqrt{17}-17} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{33-11\sqrt{17}}{-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-33+11\sqrt{17}}{8}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- \sqrt{17}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 11 } \cdot \left( 3- \sqrt{17}\right) = \color{blue}{11} \cdot3+\color{blue}{11} \cdot- \sqrt{17} = \\ = 33- 11 \sqrt{17} $$ Simplify denominator. $$ \color{blue}{ \left( 3 + \sqrt{17}\right) } \cdot \left( 3- \sqrt{17}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{17}+\color{blue}{ \sqrt{17}} \cdot3+\color{blue}{ \sqrt{17}} \cdot- \sqrt{17} = \\ = 9- 3 \sqrt{17} + 3 \sqrt{17}-17 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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