Answer
$$\frac{2}{\sqrt{5}-1}=\frac{\sqrt{5}+1}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{\sqrt{5}-1}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{\sqrt{5}-1}\frac{\sqrt{5}+1}{\sqrt{5}+1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{5}+2}{5+\sqrt{5}-\sqrt{5}-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{5}+2}{4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{5}+1}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5} + 1} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( \sqrt{5} + 1\right) = \color{blue}{2} \cdot \sqrt{5}+\color{blue}{2} \cdot1 = \\ = 2 \sqrt{5} + 2 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5}-1\right) } \cdot \left( \sqrt{5} + 1\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot1\color{blue}{-1} \cdot \sqrt{5}\color{blue}{-1} \cdot1 = \\ = 5 + \sqrt{5}- \sqrt{5}-1 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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