Question
Answer
$$\dfrac{\sqrt{3}}{-5+\sqrt{5}}=\dfrac{-5\sqrt{3}-\sqrt{15}}{20}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{3}}{-5+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{3}}{-5+\sqrt{5}}\frac{-5-\sqrt{5}}{-5-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-5\sqrt{3}-\sqrt{15}}{25+5\sqrt{5}-5\sqrt{5}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-5\sqrt{3}-\sqrt{15}}{20}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -5- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{3} } \cdot \left( -5- \sqrt{5}\right) = \color{blue}{ \sqrt{3}} \cdot-5+\color{blue}{ \sqrt{3}} \cdot- \sqrt{5} = \\ = - 5 \sqrt{3}- \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( -5 + \sqrt{5}\right) } \cdot \left( -5- \sqrt{5}\right) = \color{blue}{-5} \cdot-5\color{blue}{-5} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot-5+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 25 + 5 \sqrt{5}- 5 \sqrt{5}-5 $$ |
| ③ | Simplify numerator and denominator |
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