Answer
$$\frac{1+\sqrt{17}}{1-\sqrt{17}}=-\frac{9+\sqrt{17}}{8}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1+\sqrt{17}}{1-\sqrt{17}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1+\sqrt{17}}{1-\sqrt{17}}\frac{1+\sqrt{17}}{1+\sqrt{17}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1+\sqrt{17}+\sqrt{17}+17}{1+\sqrt{17}-\sqrt{17}-17} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18+2\sqrt{17}}{-16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{9+\sqrt{17}}{-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{9+\sqrt{17}}{8}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 1 + \sqrt{17}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 1 + \sqrt{17}\right) } \cdot \left( 1 + \sqrt{17}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot \sqrt{17}+\color{blue}{ \sqrt{17}} \cdot1+\color{blue}{ \sqrt{17}} \cdot \sqrt{17} = \\ = 1 + \sqrt{17} + \sqrt{17} + 17 $$ Simplify denominator. $$ \color{blue}{ \left( 1- \sqrt{17}\right) } \cdot \left( 1 + \sqrt{17}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot \sqrt{17}\color{blue}{- \sqrt{17}} \cdot1\color{blue}{- \sqrt{17}} \cdot \sqrt{17} = \\ = 1 + \sqrt{17}- \sqrt{17}-17 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Place a negative sign in front of a fraction. |
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