Answer
$$\frac{8}{2-\sqrt{17}}=-\frac{16+8\sqrt{17}}{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8}{2-\sqrt{17}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8}{2-\sqrt{17}}\frac{2+\sqrt{17}}{2+\sqrt{17}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{16+8\sqrt{17}}{4+2\sqrt{17}-2\sqrt{17}-17} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{16+8\sqrt{17}}{-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{16+8\sqrt{17}}{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + \sqrt{17}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 8 } \cdot \left( 2 + \sqrt{17}\right) = \color{blue}{8} \cdot2+\color{blue}{8} \cdot \sqrt{17} = \\ = 16 + 8 \sqrt{17} $$ Simplify denominator. $$ \color{blue}{ \left( 2- \sqrt{17}\right) } \cdot \left( 2 + \sqrt{17}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot \sqrt{17}\color{blue}{- \sqrt{17}} \cdot2\color{blue}{- \sqrt{17}} \cdot \sqrt{17} = \\ = 4 + 2 \sqrt{17}- 2 \sqrt{17}-17 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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