Answer
$$\frac{\sqrt{23}-\sqrt{37}}{\sqrt{23}+\sqrt{37}}=\frac{-30+\sqrt{851}}{7}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{23}-\sqrt{37}}{\sqrt{23}+\sqrt{37}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{23}-\sqrt{37}}{\sqrt{23}+\sqrt{37}}\frac{\sqrt{23}-\sqrt{37}}{\sqrt{23}-\sqrt{37}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{23-\sqrt{851}-\sqrt{851}+37}{23-\sqrt{851}+\sqrt{851}-37} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{60-2\sqrt{851}}{-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{30-\sqrt{851}}{-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-30+\sqrt{851}}{7}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{23}- \sqrt{37}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{23}- \sqrt{37}\right) } \cdot \left( \sqrt{23}- \sqrt{37}\right) = \color{blue}{ \sqrt{23}} \cdot \sqrt{23}+\color{blue}{ \sqrt{23}} \cdot- \sqrt{37}\color{blue}{- \sqrt{37}} \cdot \sqrt{23}\color{blue}{- \sqrt{37}} \cdot- \sqrt{37} = \\ = 23- \sqrt{851}- \sqrt{851} + 37 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{23} + \sqrt{37}\right) } \cdot \left( \sqrt{23}- \sqrt{37}\right) = \color{blue}{ \sqrt{23}} \cdot \sqrt{23}+\color{blue}{ \sqrt{23}} \cdot- \sqrt{37}+\color{blue}{ \sqrt{37}} \cdot \sqrt{23}+\color{blue}{ \sqrt{37}} \cdot- \sqrt{37} = \\ = 23- \sqrt{851} + \sqrt{851}-37 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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