Question
Answer
$$\dfrac{\sqrt{6}}{2-\sqrt{6}}=-\sqrt{6}-3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{6}}{2-\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{6}}{2-\sqrt{6}}\frac{2+\sqrt{6}}{2+\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{6}+6}{4+2\sqrt{6}-2\sqrt{6}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{6}+6}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{6}+3}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{\sqrt{6}+3}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }-(\sqrt{6}+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-\sqrt{6}-3\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{6} } \cdot \left( 2 + \sqrt{6}\right) = \color{blue}{ \sqrt{6}} \cdot2+\color{blue}{ \sqrt{6}} \cdot \sqrt{6} = \\ = 2 \sqrt{6} + 6 $$ Simplify denominator. $$ \color{blue}{ \left( 2- \sqrt{6}\right) } \cdot \left( 2 + \sqrt{6}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot \sqrt{6}\color{blue}{- \sqrt{6}} \cdot2\color{blue}{- \sqrt{6}} \cdot \sqrt{6} = \\ = 4 + 2 \sqrt{6}- 2 \sqrt{6}-6 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Place a negative sign in front of a fraction. |
| ⑥ | Remove the parenthesis by changing the sign of each term within them. |
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