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