Answer
$$\frac{3\sqrt{6}}{\sqrt{6}-3}=-\frac{18+9\sqrt{6}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{6}}{\sqrt{6}-3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{6}}{\sqrt{6}-3}\frac{\sqrt{6}+3}{\sqrt{6}+3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{18+9\sqrt{6}}{6+3\sqrt{6}-3\sqrt{6}-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18+9\sqrt{6}}{-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{18+9\sqrt{6}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{6} + 3} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 \sqrt{6} } \cdot \left( \sqrt{6} + 3\right) = \color{blue}{ 3 \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ 3 \sqrt{6}} \cdot3 = \\ = 18 + 9 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{6}-3\right) } \cdot \left( \sqrt{6} + 3\right) = \color{blue}{ \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot3\color{blue}{-3} \cdot \sqrt{6}\color{blue}{-3} \cdot3 = \\ = 6 + 3 \sqrt{6}- 3 \sqrt{6}-9 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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