Question
Answer
$$\dfrac{6+22\sqrt{3}}{\sqrt{3}+11}=2\sqrt{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6+22\sqrt{3}}{\sqrt{3}+11}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6+22\sqrt{3}}{\sqrt{3}+11}\frac{\sqrt{3}-11}{\sqrt{3}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{3}-66+66-242\sqrt{3}}{3-11\sqrt{3}+11\sqrt{3}-121} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-236\sqrt{3}}{-118} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{236\sqrt{3}}{118} \xlongequal{ } \\[1 em] & \xlongequal{ }2\sqrt{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{3}-11} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 6 + 22 \sqrt{3}\right) } \cdot \left( \sqrt{3}-11\right) = \color{blue}{6} \cdot \sqrt{3}+\color{blue}{6} \cdot-11+\color{blue}{ 22 \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ 22 \sqrt{3}} \cdot-11 = \\ = 6 \sqrt{3}-66 + 66- 242 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{3} + 11\right) } \cdot \left( \sqrt{3}-11\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot-11+\color{blue}{11} \cdot \sqrt{3}+\color{blue}{11} \cdot-11 = \\ = 3- 11 \sqrt{3} + 11 \sqrt{3}-121 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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