Answer
$$\frac{7}{1-\sqrt{6}}=-\frac{7+7\sqrt{6}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7}{1-\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7}{1-\sqrt{6}}\frac{1+\sqrt{6}}{1+\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{7+7\sqrt{6}}{1+\sqrt{6}-\sqrt{6}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{7+7\sqrt{6}}{-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{7+7\sqrt{6}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 1 + \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 7 } \cdot \left( 1 + \sqrt{6}\right) = \color{blue}{7} \cdot1+\color{blue}{7} \cdot \sqrt{6} = \\ = 7 + 7 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 1- \sqrt{6}\right) } \cdot \left( 1 + \sqrt{6}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot \sqrt{6}\color{blue}{- \sqrt{6}} \cdot1\color{blue}{- \sqrt{6}} \cdot \sqrt{6} = \\ = 1 + \sqrt{6}- \sqrt{6}-6 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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