Answer
$$\frac{\sqrt{2}+\sqrt{6}}{\sqrt{6}+\sqrt{8}}=\sqrt{3}-1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}+\sqrt{6}}{\sqrt{6}+\sqrt{8}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}+\sqrt{6}}{\sqrt{6}+\sqrt{8}}\frac{\sqrt{6}-\sqrt{8}}{\sqrt{6}-\sqrt{8}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{3}-4+6-4\sqrt{3}}{6-4\sqrt{3}+4\sqrt{3}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-2\sqrt{3}+2}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-\sqrt{3}+1}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{\sqrt{3}-1}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\sqrt{3}-1\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{6}- \sqrt{8}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{2} + \sqrt{6}\right) } \cdot \left( \sqrt{6}- \sqrt{8}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{6}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{8}+\color{blue}{ \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot- \sqrt{8} = \\ = 2 \sqrt{3}-4 + 6- 4 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{6} + \sqrt{8}\right) } \cdot \left( \sqrt{6}- \sqrt{8}\right) = \color{blue}{ \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot- \sqrt{8}+\color{blue}{ \sqrt{8}} \cdot \sqrt{6}+\color{blue}{ \sqrt{8}} \cdot- \sqrt{8} = \\ = 6- 4 \sqrt{3} + 4 \sqrt{3}-8 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
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