Question
Answer
$$\dfrac{4\sqrt{3}}{5\sqrt{2}-\sqrt{6}}=\dfrac{5\sqrt{6}+3\sqrt{2}}{11}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4\sqrt{3}}{5\sqrt{2}-\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4\sqrt{3}}{5\sqrt{2}-\sqrt{6}}\frac{5\sqrt{2}+\sqrt{6}}{5\sqrt{2}+\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20\sqrt{6}+12\sqrt{2}}{50+10\sqrt{3}-10\sqrt{3}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20\sqrt{6}+12\sqrt{2}}{44} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5\sqrt{6}+3\sqrt{2}}{11}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{2} + \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 \sqrt{3} } \cdot \left( 5 \sqrt{2} + \sqrt{6}\right) = \color{blue}{ 4 \sqrt{3}} \cdot 5 \sqrt{2}+\color{blue}{ 4 \sqrt{3}} \cdot \sqrt{6} = \\ = 20 \sqrt{6} + 12 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{2}- \sqrt{6}\right) } \cdot \left( 5 \sqrt{2} + \sqrt{6}\right) = \color{blue}{ 5 \sqrt{2}} \cdot 5 \sqrt{2}+\color{blue}{ 5 \sqrt{2}} \cdot \sqrt{6}\color{blue}{- \sqrt{6}} \cdot 5 \sqrt{2}\color{blue}{- \sqrt{6}} \cdot \sqrt{6} = \\ = 50 + 10 \sqrt{3}- 10 \sqrt{3}-6 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
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