Answer
$$\frac{4}{-1+\sqrt{14}}=\frac{4+4\sqrt{14}}{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{-1+\sqrt{14}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{-1+\sqrt{14}}\frac{-1-\sqrt{14}}{-1-\sqrt{14}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-4-4\sqrt{14}}{1+\sqrt{14}-\sqrt{14}-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-4-4\sqrt{14}}{-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4+4\sqrt{14}}{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -1- \sqrt{14}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( -1- \sqrt{14}\right) = \color{blue}{4} \cdot-1+\color{blue}{4} \cdot- \sqrt{14} = \\ = -4- 4 \sqrt{14} $$ Simplify denominator. $$ \color{blue}{ \left( -1 + \sqrt{14}\right) } \cdot \left( -1- \sqrt{14}\right) = \color{blue}{-1} \cdot-1\color{blue}{-1} \cdot- \sqrt{14}+\color{blue}{ \sqrt{14}} \cdot-1+\color{blue}{ \sqrt{14}} \cdot- \sqrt{14} = \\ = 1 + \sqrt{14}- \sqrt{14}-14 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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