Question
Answer
$$\dfrac{\sqrt{2}-11}{\sqrt{7}+6}=\dfrac{-\sqrt{14}+6\sqrt{2}+11\sqrt{7}-66}{29}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}-11}{\sqrt{7}+6}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}-11}{\sqrt{7}+6}\frac{\sqrt{7}-6}{\sqrt{7}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{14}-6\sqrt{2}-11\sqrt{7}+66}{7-6\sqrt{7}+6\sqrt{7}-36} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{14}-6\sqrt{2}-11\sqrt{7}+66}{-29} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-\sqrt{14}+6\sqrt{2}+11\sqrt{7}-66}{29}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{7}-6} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{2}-11\right) } \cdot \left( \sqrt{7}-6\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{7}+\color{blue}{ \sqrt{2}} \cdot-6\color{blue}{-11} \cdot \sqrt{7}\color{blue}{-11} \cdot-6 = \\ = \sqrt{14}- 6 \sqrt{2}- 11 \sqrt{7} + 66 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{7} + 6\right) } \cdot \left( \sqrt{7}-6\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot-6+\color{blue}{6} \cdot \sqrt{7}+\color{blue}{6} \cdot-6 = \\ = 7- 6 \sqrt{7} + 6 \sqrt{7}-36 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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