Answer
$$\frac{4}{\sqrt{5}-7}=-\frac{\sqrt{5}+7}{11}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{\sqrt{5}-7}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{\sqrt{5}-7}\frac{\sqrt{5}+7}{\sqrt{5}+7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4\sqrt{5}+28}{5+7\sqrt{5}-7\sqrt{5}-49} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4\sqrt{5}+28}{-44} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{5}+7}{-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{\sqrt{5}+7}{11}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5} + 7} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( \sqrt{5} + 7\right) = \color{blue}{4} \cdot \sqrt{5}+\color{blue}{4} \cdot7 = \\ = 4 \sqrt{5} + 28 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5}-7\right) } \cdot \left( \sqrt{5} + 7\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot7\color{blue}{-7} \cdot \sqrt{5}\color{blue}{-7} \cdot7 = \\ = 5 + 7 \sqrt{5}- 7 \sqrt{5}-49 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
| ⑤ | Place a negative sign in front of a fraction. |
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