Question
Answer
$$\dfrac{5-\sqrt{37}}{5+\sqrt{37}}=\dfrac{-31+5\sqrt{37}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5-\sqrt{37}}{5+\sqrt{37}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5-\sqrt{37}}{5+\sqrt{37}}\frac{5-\sqrt{37}}{5-\sqrt{37}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{25-5\sqrt{37}-5\sqrt{37}+37}{25-5\sqrt{37}+5\sqrt{37}-37} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{62-10\sqrt{37}}{-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{31-5\sqrt{37}}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-31+5\sqrt{37}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- \sqrt{37}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5- \sqrt{37}\right) } \cdot \left( 5- \sqrt{37}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- \sqrt{37}\color{blue}{- \sqrt{37}} \cdot5\color{blue}{- \sqrt{37}} \cdot- \sqrt{37} = \\ = 25- 5 \sqrt{37}- 5 \sqrt{37} + 37 $$ Simplify denominator. $$ \color{blue}{ \left( 5 + \sqrt{37}\right) } \cdot \left( 5- \sqrt{37}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- \sqrt{37}+\color{blue}{ \sqrt{37}} \cdot5+\color{blue}{ \sqrt{37}} \cdot- \sqrt{37} = \\ = 25- 5 \sqrt{37} + 5 \sqrt{37}-37 $$ |
| 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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