Question
Answer
$$\dfrac{\sqrt{3}+\sqrt{13}}{\sqrt{3}-\sqrt{13}}=-\dfrac{8+\sqrt{39}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{3}+\sqrt{13}}{\sqrt{3}-\sqrt{13}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{3}+\sqrt{13}}{\sqrt{3}-\sqrt{13}}\frac{\sqrt{3}+\sqrt{13}}{\sqrt{3}+\sqrt{13}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3+\sqrt{39}+\sqrt{39}+13}{3+\sqrt{39}-\sqrt{39}-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{16+2\sqrt{39}}{-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{8+\sqrt{39}}{-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{8+\sqrt{39}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{3} + \sqrt{13}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{3} + \sqrt{13}\right) } \cdot \left( \sqrt{3} + \sqrt{13}\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot \sqrt{13}+\color{blue}{ \sqrt{13}} \cdot \sqrt{3}+\color{blue}{ \sqrt{13}} \cdot \sqrt{13} = \\ = 3 + \sqrt{39} + \sqrt{39} + 13 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{3}- \sqrt{13}\right) } \cdot \left( \sqrt{3} + \sqrt{13}\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot \sqrt{13}\color{blue}{- \sqrt{13}} \cdot \sqrt{3}\color{blue}{- \sqrt{13}} \cdot \sqrt{13} = \\ = 3 + \sqrt{39}- \sqrt{39}-13 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Place a negative sign in front of a fraction. |
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