Answer
$$\frac{\sqrt{21}+7}{\sqrt{21}-7}=-\frac{5+\sqrt{21}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{21}+7}{\sqrt{21}-7}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{21}+7}{\sqrt{21}-7}\frac{\sqrt{21}+7}{\sqrt{21}+7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{21+7\sqrt{21}+7\sqrt{21}+49}{21+7\sqrt{21}-7\sqrt{21}-49} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{70+14\sqrt{21}}{-28} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5+\sqrt{21}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{5+\sqrt{21}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{21} + 7} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{21} + 7\right) } \cdot \left( \sqrt{21} + 7\right) = \color{blue}{ \sqrt{21}} \cdot \sqrt{21}+\color{blue}{ \sqrt{21}} \cdot7+\color{blue}{7} \cdot \sqrt{21}+\color{blue}{7} \cdot7 = \\ = 21 + 7 \sqrt{21} + 7 \sqrt{21} + 49 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{21}-7\right) } \cdot \left( \sqrt{21} + 7\right) = \color{blue}{ \sqrt{21}} \cdot \sqrt{21}+\color{blue}{ \sqrt{21}} \cdot7\color{blue}{-7} \cdot \sqrt{21}\color{blue}{-7} \cdot7 = \\ = 21 + 7 \sqrt{21}- 7 \sqrt{21}-49 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 14. |
| ⑤ | Place a negative sign in front of a fraction. |
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