Answer
$$\frac{6}{\sqrt{18}-\sqrt{12}}=3\sqrt{2}+2\sqrt{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6}{\sqrt{18}-\sqrt{12}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6}{\sqrt{18}-\sqrt{12}}\frac{\sqrt{18}+\sqrt{12}}{\sqrt{18}+\sqrt{12}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{18\sqrt{2}+12\sqrt{3}}{18+6\sqrt{6}-6\sqrt{6}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18\sqrt{2}+12\sqrt{3}}{6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3\sqrt{2}+2\sqrt{3}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}3\sqrt{2}+2\sqrt{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{18} + \sqrt{12}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 } \cdot \left( \sqrt{18} + \sqrt{12}\right) = \color{blue}{6} \cdot \sqrt{18}+\color{blue}{6} \cdot \sqrt{12} = \\ = 18 \sqrt{2} + 12 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{18}- \sqrt{12}\right) } \cdot \left( \sqrt{18} + \sqrt{12}\right) = \color{blue}{ \sqrt{18}} \cdot \sqrt{18}+\color{blue}{ \sqrt{18}} \cdot \sqrt{12}\color{blue}{- \sqrt{12}} \cdot \sqrt{18}\color{blue}{- \sqrt{12}} \cdot \sqrt{12} = \\ = 18 + 6 \sqrt{6}- 6 \sqrt{6}-12 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 6. |
| ⑤ | Remove 1 from denominator. |
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