Question
Answer
$$\dfrac{\sqrt{2}}{-\sqrt{5}+\sqrt{15}}=\dfrac{\sqrt{10}+\sqrt{30}}{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{-\sqrt{5}+\sqrt{15}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{-\sqrt{5}+\sqrt{15}}\frac{-\sqrt{5}-\sqrt{15}}{-\sqrt{5}-\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-\sqrt{10}-\sqrt{30}}{5+5\sqrt{3}-5\sqrt{3}-15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-\sqrt{10}-\sqrt{30}}{-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{10}+\sqrt{30}}{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ - \sqrt{5}- \sqrt{15}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( - \sqrt{5}- \sqrt{15}\right) = \color{blue}{ \sqrt{2}} \cdot- \sqrt{5}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{15} = \\ = - \sqrt{10}- \sqrt{30} $$ Simplify denominator. $$ \color{blue}{ \left( - \sqrt{5} + \sqrt{15}\right) } \cdot \left( - \sqrt{5}- \sqrt{15}\right) = \color{blue}{- \sqrt{5}} \cdot- \sqrt{5}\color{blue}{- \sqrt{5}} \cdot- \sqrt{15}+\color{blue}{ \sqrt{15}} \cdot- \sqrt{5}+\color{blue}{ \sqrt{15}} \cdot- \sqrt{15} = \\ = 5 + 5 \sqrt{3}- 5 \sqrt{3}-15 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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