Question
Answer
$$\dfrac{5\sqrt{3}}{4-\sqrt{11}}=4\sqrt{3}+\sqrt{33}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{3}}{4-\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{3}}{4-\sqrt{11}}\frac{4+\sqrt{11}}{4+\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20\sqrt{3}+5\sqrt{33}}{16+4\sqrt{11}-4\sqrt{11}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20\sqrt{3}+5\sqrt{33}}{5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4\sqrt{3}+\sqrt{33}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4\sqrt{3}+\sqrt{33}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 + \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 \sqrt{3} } \cdot \left( 4 + \sqrt{11}\right) = \color{blue}{ 5 \sqrt{3}} \cdot4+\color{blue}{ 5 \sqrt{3}} \cdot \sqrt{11} = \\ = 20 \sqrt{3} + 5 \sqrt{33} $$ Simplify denominator. $$ \color{blue}{ \left( 4- \sqrt{11}\right) } \cdot \left( 4 + \sqrt{11}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot \sqrt{11}\color{blue}{- \sqrt{11}} \cdot4\color{blue}{- \sqrt{11}} \cdot \sqrt{11} = \\ = 16 + 4 \sqrt{11}- 4 \sqrt{11}-11 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
| ⑤ | Remove 1 from denominator. |
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