Question
Answer
$$\dfrac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}=12+\sqrt{143}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}+\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{13+\sqrt{143}+\sqrt{143}+11}{13+\sqrt{143}-\sqrt{143}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{24+2\sqrt{143}}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{12+\sqrt{143}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}12+\sqrt{143}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{13} + \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{13} + \sqrt{11}\right) } \cdot \left( \sqrt{13} + \sqrt{11}\right) = \color{blue}{ \sqrt{13}} \cdot \sqrt{13}+\color{blue}{ \sqrt{13}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot \sqrt{13}+\color{blue}{ \sqrt{11}} \cdot \sqrt{11} = \\ = 13 + \sqrt{143} + \sqrt{143} + 11 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{13}- \sqrt{11}\right) } \cdot \left( \sqrt{13} + \sqrt{11}\right) = \color{blue}{ \sqrt{13}} \cdot \sqrt{13}+\color{blue}{ \sqrt{13}} \cdot \sqrt{11}\color{blue}{- \sqrt{11}} \cdot \sqrt{13}\color{blue}{- \sqrt{11}} \cdot \sqrt{11} = \\ = 13 + \sqrt{143}- \sqrt{143}-11 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Remove 1 from denominator. |
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