Answer
$$\frac{6\sqrt{6}}{2\sqrt{3}+4}=\frac{-36\sqrt{2}+24\sqrt{6}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6\sqrt{6}}{2\sqrt{3}+4}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6\sqrt{6}}{2\sqrt{3}+4}\frac{2\sqrt{3}-4}{2\sqrt{3}-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{36\sqrt{2}-24\sqrt{6}}{12-8\sqrt{3}+8\sqrt{3}-16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{36\sqrt{2}-24\sqrt{6}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-36\sqrt{2}+24\sqrt{6}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{3}-4} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 \sqrt{6} } \cdot \left( 2 \sqrt{3}-4\right) = \color{blue}{ 6 \sqrt{6}} \cdot 2 \sqrt{3}+\color{blue}{ 6 \sqrt{6}} \cdot-4 = \\ = 36 \sqrt{2}- 24 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{3} + 4\right) } \cdot \left( 2 \sqrt{3}-4\right) = \color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot-4+\color{blue}{4} \cdot 2 \sqrt{3}+\color{blue}{4} \cdot-4 = \\ = 12- 8 \sqrt{3} + 8 \sqrt{3}-16 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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