Answer
$$\frac{17}{2+\sqrt{23}}=\frac{-34+17\sqrt{23}}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{17}{2+\sqrt{23}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{17}{2+\sqrt{23}}\frac{2-\sqrt{23}}{2-\sqrt{23}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{34-17\sqrt{23}}{4-2\sqrt{23}+2\sqrt{23}-23} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{34-17\sqrt{23}}{-19} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-34+17\sqrt{23}}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2- \sqrt{23}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 17 } \cdot \left( 2- \sqrt{23}\right) = \color{blue}{17} \cdot2+\color{blue}{17} \cdot- \sqrt{23} = \\ = 34- 17 \sqrt{23} $$ Simplify denominator. $$ \color{blue}{ \left( 2 + \sqrt{23}\right) } \cdot \left( 2- \sqrt{23}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- \sqrt{23}+\color{blue}{ \sqrt{23}} \cdot2+\color{blue}{ \sqrt{23}} \cdot- \sqrt{23} = \\ = 4- 2 \sqrt{23} + 2 \sqrt{23}-23 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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