Answer
$$\frac{\sqrt{2}}{\sqrt{12}-\sqrt{8}}=\frac{\sqrt{6}+2}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{\sqrt{12}-\sqrt{8}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{\sqrt{12}-\sqrt{8}}\frac{\sqrt{12}+\sqrt{8}}{\sqrt{12}+\sqrt{8}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{6}+4}{12+4\sqrt{6}-4\sqrt{6}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{6}+4}{4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{6}+2}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{12} + \sqrt{8}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( \sqrt{12} + \sqrt{8}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{12}+\color{blue}{ \sqrt{2}} \cdot \sqrt{8} = \\ = 2 \sqrt{6} + 4 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{12}- \sqrt{8}\right) } \cdot \left( \sqrt{12} + \sqrt{8}\right) = \color{blue}{ \sqrt{12}} \cdot \sqrt{12}+\color{blue}{ \sqrt{12}} \cdot \sqrt{8}\color{blue}{- \sqrt{8}} \cdot \sqrt{12}\color{blue}{- \sqrt{8}} \cdot \sqrt{8} = \\ = 12 + 4 \sqrt{6}- 4 \sqrt{6}-8 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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