Answer
$$\frac{2\sqrt{2}}{\sqrt{16}-\sqrt{12}}=2\sqrt{2}+\sqrt{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{2}}{\sqrt{16}-\sqrt{12}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{2}}{\sqrt{16}-\sqrt{12}}\frac{\sqrt{16}+\sqrt{12}}{\sqrt{16}+\sqrt{12}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8\sqrt{2}+4\sqrt{6}}{16+8\sqrt{3}-8\sqrt{3}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8\sqrt{2}+4\sqrt{6}}{4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2\sqrt{2}+\sqrt{6}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}2\sqrt{2}+\sqrt{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{16} + \sqrt{12}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 \sqrt{2} } \cdot \left( \sqrt{16} + \sqrt{12}\right) = \color{blue}{ 2 \sqrt{2}} \cdot \sqrt{16}+\color{blue}{ 2 \sqrt{2}} \cdot \sqrt{12} = \\ = 8 \sqrt{2} + 4 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{16}- \sqrt{12}\right) } \cdot \left( \sqrt{16} + \sqrt{12}\right) = \color{blue}{ \sqrt{16}} \cdot \sqrt{16}+\color{blue}{ \sqrt{16}} \cdot \sqrt{12}\color{blue}{- \sqrt{12}} \cdot \sqrt{16}\color{blue}{- \sqrt{12}} \cdot \sqrt{12} = \\ = 16 + 8 \sqrt{3}- 8 \sqrt{3}-12 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
| ⑤ | Remove 1 from denominator. |
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