Answer
$$\frac{4-\sqrt{14}}{2+\sqrt{14}}=\frac{-11+3\sqrt{14}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4-\sqrt{14}}{2+\sqrt{14}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4-\sqrt{14}}{2+\sqrt{14}}\frac{2-\sqrt{14}}{2-\sqrt{14}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8-4\sqrt{14}-2\sqrt{14}+14}{4-2\sqrt{14}+2\sqrt{14}-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{22-6\sqrt{14}}{-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{11-3\sqrt{14}}{-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-11+3\sqrt{14}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2- \sqrt{14}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4- \sqrt{14}\right) } \cdot \left( 2- \sqrt{14}\right) = \color{blue}{4} \cdot2+\color{blue}{4} \cdot- \sqrt{14}\color{blue}{- \sqrt{14}} \cdot2\color{blue}{- \sqrt{14}} \cdot- \sqrt{14} = \\ = 8- 4 \sqrt{14}- 2 \sqrt{14} + 14 $$ Simplify denominator. $$ \color{blue}{ \left( 2 + \sqrt{14}\right) } \cdot \left( 2- \sqrt{14}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- \sqrt{14}+\color{blue}{ \sqrt{14}} \cdot2+\color{blue}{ \sqrt{14}} \cdot- \sqrt{14} = \\ = 4- 2 \sqrt{14} + 2 \sqrt{14}-14 $$ |
| 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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