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