Question
Answer
$$\dfrac{7}{9-\sqrt{11}}=\dfrac{9+\sqrt{11}}{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7}{9-\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7}{9-\sqrt{11}}\frac{9+\sqrt{11}}{9+\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{63+7\sqrt{11}}{81+9\sqrt{11}-9\sqrt{11}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{63+7\sqrt{11}}{70} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{9+\sqrt{11}}{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 9 + \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 7 } \cdot \left( 9 + \sqrt{11}\right) = \color{blue}{7} \cdot9+\color{blue}{7} \cdot \sqrt{11} = \\ = 63 + 7 \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ \left( 9- \sqrt{11}\right) } \cdot \left( 9 + \sqrt{11}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot \sqrt{11}\color{blue}{- \sqrt{11}} \cdot9\color{blue}{- \sqrt{11}} \cdot \sqrt{11} = \\ = 81 + 9 \sqrt{11}- 9 \sqrt{11}-11 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 7. |
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