Answer
$$\frac{2-\sqrt{14}}{3\sqrt{6}+8}=\frac{-3\sqrt{6}+8+3\sqrt{21}-4\sqrt{14}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2-\sqrt{14}}{3\sqrt{6}+8}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2-\sqrt{14}}{3\sqrt{6}+8}\frac{3\sqrt{6}-8}{3\sqrt{6}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{6}-16-6\sqrt{21}+8\sqrt{14}}{54-24\sqrt{6}+24\sqrt{6}-64} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6\sqrt{6}-16-6\sqrt{21}+8\sqrt{14}}{-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3\sqrt{6}-8-3\sqrt{21}+4\sqrt{14}}{-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-3\sqrt{6}+8+3\sqrt{21}-4\sqrt{14}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{6}-8} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2- \sqrt{14}\right) } \cdot \left( 3 \sqrt{6}-8\right) = \color{blue}{2} \cdot 3 \sqrt{6}+\color{blue}{2} \cdot-8\color{blue}{- \sqrt{14}} \cdot 3 \sqrt{6}\color{blue}{- \sqrt{14}} \cdot-8 = \\ = 6 \sqrt{6}-16- 6 \sqrt{21} + 8 \sqrt{14} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{6} + 8\right) } \cdot \left( 3 \sqrt{6}-8\right) = \color{blue}{ 3 \sqrt{6}} \cdot 3 \sqrt{6}+\color{blue}{ 3 \sqrt{6}} \cdot-8+\color{blue}{8} \cdot 3 \sqrt{6}+\color{blue}{8} \cdot-8 = \\ = 54- 24 \sqrt{6} + 24 \sqrt{6}-64 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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