Answer
$$\frac{4}{\sqrt{2}-\sqrt{6}}=-\sqrt{2}-\sqrt{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{\sqrt{2}-\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{\sqrt{2}-\sqrt{6}}\frac{\sqrt{2}+\sqrt{6}}{\sqrt{2}+\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4\sqrt{2}+4\sqrt{6}}{2+2\sqrt{3}-2\sqrt{3}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4\sqrt{2}+4\sqrt{6}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{2}+\sqrt{6}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{\sqrt{2}+\sqrt{6}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }-(\sqrt{2}+\sqrt{6}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-\sqrt{2}-\sqrt{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{2} + \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( \sqrt{2} + \sqrt{6}\right) = \color{blue}{4} \cdot \sqrt{2}+\color{blue}{4} \cdot \sqrt{6} = \\ = 4 \sqrt{2} + 4 \sqrt{6} $$ 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 | Divide both numerator and denominator by 4. |
| ⑤ | Place a negative sign in front of a fraction. |
| ⑥ | Remove the parenthesis by changing the sign of each term within them. |
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