Question
Answer
$$\dfrac{1}{2\sqrt{3}-\sqrt{11}}=2\sqrt{3}+\sqrt{11}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{2\sqrt{3}-\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{2\sqrt{3}-\sqrt{11}}\frac{2\sqrt{3}+\sqrt{11}}{2\sqrt{3}+\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{3}+\sqrt{11}}{12+2\sqrt{33}-2\sqrt{33}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{3}+\sqrt{11}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2\sqrt{3}+\sqrt{11}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{3} + \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 2 \sqrt{3} + \sqrt{11}\right) = \color{blue}{1} \cdot 2 \sqrt{3}+\color{blue}{1} \cdot \sqrt{11} = \\ = 2 \sqrt{3} + \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{3}- \sqrt{11}\right) } \cdot \left( 2 \sqrt{3} + \sqrt{11}\right) = \color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot \sqrt{11}\color{blue}{- \sqrt{11}} \cdot 2 \sqrt{3}\color{blue}{- \sqrt{11}} \cdot \sqrt{11} = \\ = 12 + 2 \sqrt{33}- 2 \sqrt{33}-11 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Remove 1 from denominator. |
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