Answer
$$\frac{8}{1-\sqrt{14}}=-\frac{8+8\sqrt{14}}{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8}{1-\sqrt{14}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8}{1-\sqrt{14}}\frac{1+\sqrt{14}}{1+\sqrt{14}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8+8\sqrt{14}}{1+\sqrt{14}-\sqrt{14}-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8+8\sqrt{14}}{-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{8+8\sqrt{14}}{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 1 + \sqrt{14}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 8 } \cdot \left( 1 + \sqrt{14}\right) = \color{blue}{8} \cdot1+\color{blue}{8} \cdot \sqrt{14} = \\ = 8 + 8 \sqrt{14} $$ Simplify denominator. $$ \color{blue}{ \left( 1- \sqrt{14}\right) } \cdot \left( 1 + \sqrt{14}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot \sqrt{14}\color{blue}{- \sqrt{14}} \cdot1\color{blue}{- \sqrt{14}} \cdot \sqrt{14} = \\ = 1 + \sqrt{14}- \sqrt{14}-14 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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