Answer
$$\frac{5+\sqrt{3}}{\sqrt{2}+4}=\frac{-5\sqrt{2}+20-\sqrt{6}+4\sqrt{3}}{14}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5+\sqrt{3}}{\sqrt{2}+4}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5+\sqrt{3}}{\sqrt{2}+4}\frac{\sqrt{2}-4}{\sqrt{2}-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5\sqrt{2}-20+\sqrt{6}-4\sqrt{3}}{2-4\sqrt{2}+4\sqrt{2}-16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5\sqrt{2}-20+\sqrt{6}-4\sqrt{3}}{-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-5\sqrt{2}+20-\sqrt{6}+4\sqrt{3}}{14}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{2}-4} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 + \sqrt{3}\right) } \cdot \left( \sqrt{2}-4\right) = \color{blue}{5} \cdot \sqrt{2}+\color{blue}{5} \cdot-4+\color{blue}{ \sqrt{3}} \cdot \sqrt{2}+\color{blue}{ \sqrt{3}} \cdot-4 = \\ = 5 \sqrt{2}-20 + \sqrt{6}- 4 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{2} + 4\right) } \cdot \left( \sqrt{2}-4\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot-4+\color{blue}{4} \cdot \sqrt{2}+\color{blue}{4} \cdot-4 = \\ = 2- 4 \sqrt{2} + 4 \sqrt{2}-16 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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