Answer
$$\frac{2+8\sqrt{6}}{4+\sqrt{6}}=-4+3\sqrt{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2+8\sqrt{6}}{4+\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2+8\sqrt{6}}{4+\sqrt{6}}\frac{4-\sqrt{6}}{4-\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8-2\sqrt{6}+32\sqrt{6}-48}{16-4\sqrt{6}+4\sqrt{6}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-40+30\sqrt{6}}{10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-4+3\sqrt{6}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-4+3\sqrt{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4- \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 + 8 \sqrt{6}\right) } \cdot \left( 4- \sqrt{6}\right) = \color{blue}{2} \cdot4+\color{blue}{2} \cdot- \sqrt{6}+\color{blue}{ 8 \sqrt{6}} \cdot4+\color{blue}{ 8 \sqrt{6}} \cdot- \sqrt{6} = \\ = 8- 2 \sqrt{6} + 32 \sqrt{6}-48 $$ Simplify denominator. $$ \color{blue}{ \left( 4 + \sqrt{6}\right) } \cdot \left( 4- \sqrt{6}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot- \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot4+\color{blue}{ \sqrt{6}} \cdot- \sqrt{6} = \\ = 16- 4 \sqrt{6} + 4 \sqrt{6}-6 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 10. |
| ⑤ | Remove 1 from denominator. |
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