Question
Answer
$$\dfrac{9+\sqrt{5}}{\sqrt{7}+10}=\dfrac{-9\sqrt{7}+90-\sqrt{35}+10\sqrt{5}}{93}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{9+\sqrt{5}}{\sqrt{7}+10}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{9+\sqrt{5}}{\sqrt{7}+10}\frac{\sqrt{7}-10}{\sqrt{7}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9\sqrt{7}-90+\sqrt{35}-10\sqrt{5}}{7-10\sqrt{7}+10\sqrt{7}-100} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9\sqrt{7}-90+\sqrt{35}-10\sqrt{5}}{-93} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-9\sqrt{7}+90-\sqrt{35}+10\sqrt{5}}{93}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{7}-10} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 9 + \sqrt{5}\right) } \cdot \left( \sqrt{7}-10\right) = \color{blue}{9} \cdot \sqrt{7}+\color{blue}{9} \cdot-10+\color{blue}{ \sqrt{5}} \cdot \sqrt{7}+\color{blue}{ \sqrt{5}} \cdot-10 = \\ = 9 \sqrt{7}-90 + \sqrt{35}- 10 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{7} + 10\right) } \cdot \left( \sqrt{7}-10\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot-10+\color{blue}{10} \cdot \sqrt{7}+\color{blue}{10} \cdot-10 = \\ = 7- 10 \sqrt{7} + 10 \sqrt{7}-100 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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