Question
Answer
$$\dfrac{14+\sqrt{7}}{4\sqrt{7}+11}=-5\sqrt{7}+14$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{14+\sqrt{7}}{4\sqrt{7}+11}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{14+\sqrt{7}}{4\sqrt{7}+11}\frac{4\sqrt{7}-11}{4\sqrt{7}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{56\sqrt{7}-154+28-11\sqrt{7}}{112-44\sqrt{7}+44\sqrt{7}-121} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{45\sqrt{7}-126}{-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5\sqrt{7}-14}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-5\sqrt{7}+14}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-5\sqrt{7}+14\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{7}-11} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 14 + \sqrt{7}\right) } \cdot \left( 4 \sqrt{7}-11\right) = \color{blue}{14} \cdot 4 \sqrt{7}+\color{blue}{14} \cdot-11+\color{blue}{ \sqrt{7}} \cdot 4 \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot-11 = \\ = 56 \sqrt{7}-154 + 28- 11 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{7} + 11\right) } \cdot \left( 4 \sqrt{7}-11\right) = \color{blue}{ 4 \sqrt{7}} \cdot 4 \sqrt{7}+\color{blue}{ 4 \sqrt{7}} \cdot-11+\color{blue}{11} \cdot 4 \sqrt{7}+\color{blue}{11} \cdot-11 = \\ = 112- 44 \sqrt{7} + 44 \sqrt{7}-121 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 9. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
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