Question
Answer
$$\dfrac{2\sqrt{6}+6\sqrt{2}-8\sqrt{3}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{3}-\sqrt{6}+\sqrt{2}}=-2\sqrt{3}+3\sqrt{2}-18+7\sqrt{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{6}+6\sqrt{2}-8\sqrt{3}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{3}-\sqrt{6}+\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{6}+6\sqrt{2}-8\sqrt{3}}{2\sqrt{2}+2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{6}+6\sqrt{2}-8\sqrt{3}}{2\sqrt{2}+2\sqrt{3}}\frac{2\sqrt{2}-2\sqrt{3}}{2\sqrt{2}-2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8\sqrt{3}-12\sqrt{2}+24-12\sqrt{6}-16\sqrt{6}+48}{8-4\sqrt{6}+4\sqrt{6}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{8\sqrt{3}-12\sqrt{2}+72-28\sqrt{6}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{2\sqrt{3}-3\sqrt{2}+18-7\sqrt{6}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{-2\sqrt{3}+3\sqrt{2}-18+7\sqrt{6}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}-2\sqrt{3}+3\sqrt{2}-18+7\sqrt{6}\end{aligned} $$ | |
| ① | Simplify numerator and denominator |
| ② | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{2}- 2 \sqrt{3}} $$. |
| ③ | Multiply in a numerator. $$ \color{blue}{ \left( 2 \sqrt{6} + 6 \sqrt{2}- 8 \sqrt{3}\right) } \cdot \left( 2 \sqrt{2}- 2 \sqrt{3}\right) = \color{blue}{ 2 \sqrt{6}} \cdot 2 \sqrt{2}+\color{blue}{ 2 \sqrt{6}} \cdot- 2 \sqrt{3}+\color{blue}{ 6 \sqrt{2}} \cdot 2 \sqrt{2}+\color{blue}{ 6 \sqrt{2}} \cdot- 2 \sqrt{3}\color{blue}{- 8 \sqrt{3}} \cdot 2 \sqrt{2}\color{blue}{- 8 \sqrt{3}} \cdot- 2 \sqrt{3} = \\ = 8 \sqrt{3}- 12 \sqrt{2} + 24- 12 \sqrt{6}- 16 \sqrt{6} + 48 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{2} + 2 \sqrt{3}\right) } \cdot \left( 2 \sqrt{2}- 2 \sqrt{3}\right) = \color{blue}{ 2 \sqrt{2}} \cdot 2 \sqrt{2}+\color{blue}{ 2 \sqrt{2}} \cdot- 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{2}+\color{blue}{ 2 \sqrt{3}} \cdot- 2 \sqrt{3} = \\ = 8- 4 \sqrt{6} + 4 \sqrt{6}-12 $$ |
| ④ | Simplify numerator and denominator |
| ⑤ | Divide both numerator and denominator by 4. |
| ⑥ | Multiply both numerator and denominator by -1. |
| ⑦ | Remove 1 from denominator. |
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