Question
Answer
$$\dfrac{10}{-\sqrt{2}+\sqrt{6}}=\dfrac{10\sqrt{2}+10\sqrt{6}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{10}{-\sqrt{2}+\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{10}{-\sqrt{2}+\sqrt{6}}\frac{-\sqrt{2}-\sqrt{6}}{-\sqrt{2}-\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-10\sqrt{2}-10\sqrt{6}}{2+2\sqrt{3}-2\sqrt{3}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-10\sqrt{2}-10\sqrt{6}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{10\sqrt{2}+10\sqrt{6}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ - \sqrt{2}- \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 10 } \cdot \left( - \sqrt{2}- \sqrt{6}\right) = \color{blue}{10} \cdot- \sqrt{2}+\color{blue}{10} \cdot- \sqrt{6} = \\ = - 10 \sqrt{2}- 10 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( - \sqrt{2} + \sqrt{6}\right) } \cdot \left( - \sqrt{2}- \sqrt{6}\right) = \color{blue}{- \sqrt{2}} \cdot- \sqrt{2}\color{blue}{- \sqrt{2}} \cdot- \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot- \sqrt{2}+\color{blue}{ \sqrt{6}} \cdot- \sqrt{6} = \\ = 2 + 2 \sqrt{3}- 2 \sqrt{3}-6 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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