Answer
$$\frac{\sqrt{5}+1}{\sqrt{5}-3}=-(2+\sqrt{5})$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}+1}{\sqrt{5}-3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}+1}{\sqrt{5}-3}\frac{\sqrt{5}+3}{\sqrt{5}+3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5+3\sqrt{5}+\sqrt{5}+3}{5+3\sqrt{5}-3\sqrt{5}-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8+4\sqrt{5}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2+\sqrt{5}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{2+\sqrt{5}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }-(2+\sqrt{5})\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5} + 3} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{5} + 1\right) } \cdot \left( \sqrt{5} + 3\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot3+\color{blue}{1} \cdot \sqrt{5}+\color{blue}{1} \cdot3 = \\ = 5 + 3 \sqrt{5} + \sqrt{5} + 3 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5}-3\right) } \cdot \left( \sqrt{5} + 3\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot3\color{blue}{-3} \cdot \sqrt{5}\color{blue}{-3} \cdot3 = \\ = 5 + 3 \sqrt{5}- 3 \sqrt{5}-9 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
| ⑤ | Place a negative sign in front of a fraction. |
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