Answer
$$\frac{\sqrt{4}-4}{\sqrt{14}-9}=\frac{2\sqrt{14}+18}{67}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{4}-4}{\sqrt{14}-9}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{4}-4}{\sqrt{14}-9}\frac{\sqrt{14}+9}{\sqrt{14}+9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{14}+18-4\sqrt{14}-36}{14+9\sqrt{14}-9\sqrt{14}-81} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-2\sqrt{14}-18}{-67} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2\sqrt{14}+18}{67}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{14} + 9} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{4}-4\right) } \cdot \left( \sqrt{14} + 9\right) = \color{blue}{ \sqrt{4}} \cdot \sqrt{14}+\color{blue}{ \sqrt{4}} \cdot9\color{blue}{-4} \cdot \sqrt{14}\color{blue}{-4} \cdot9 = \\ = 2 \sqrt{14} + 18- 4 \sqrt{14}-36 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{14}-9\right) } \cdot \left( \sqrt{14} + 9\right) = \color{blue}{ \sqrt{14}} \cdot \sqrt{14}+\color{blue}{ \sqrt{14}} \cdot9\color{blue}{-9} \cdot \sqrt{14}\color{blue}{-9} \cdot9 = \\ = 14 + 9 \sqrt{14}- 9 \sqrt{14}-81 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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