Question
Answer
$$\dfrac{11-\sqrt{5}}{\sqrt{5}+7}=\dfrac{-9\sqrt{5}+41}{22}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{11-\sqrt{5}}{\sqrt{5}+7}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{11-\sqrt{5}}{\sqrt{5}+7}\frac{\sqrt{5}-7}{\sqrt{5}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{11\sqrt{5}-77-5+7\sqrt{5}}{5-7\sqrt{5}+7\sqrt{5}-49} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18\sqrt{5}-82}{-44} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{9\sqrt{5}-41}{-22} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-9\sqrt{5}+41}{22}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5}-7} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 11- \sqrt{5}\right) } \cdot \left( \sqrt{5}-7\right) = \color{blue}{11} \cdot \sqrt{5}+\color{blue}{11} \cdot-7\color{blue}{- \sqrt{5}} \cdot \sqrt{5}\color{blue}{- \sqrt{5}} \cdot-7 = \\ = 11 \sqrt{5}-77-5 + 7 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5} + 7\right) } \cdot \left( \sqrt{5}-7\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot-7+\color{blue}{7} \cdot \sqrt{5}+\color{blue}{7} \cdot-7 = \\ = 5- 7 \sqrt{5} + 7 \sqrt{5}-49 $$ |
| 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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