Answer
$$\frac{5\sqrt{2}}{3\sqrt{6}-2}=\frac{3\sqrt{3}+\sqrt{2}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{2}}{3\sqrt{6}-2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{2}}{3\sqrt{6}-2}\frac{3\sqrt{6}+2}{3\sqrt{6}+2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30\sqrt{3}+10\sqrt{2}}{54+6\sqrt{6}-6\sqrt{6}-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{30\sqrt{3}+10\sqrt{2}}{50} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3\sqrt{3}+\sqrt{2}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{6} + 2} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 \sqrt{2} } \cdot \left( 3 \sqrt{6} + 2\right) = \color{blue}{ 5 \sqrt{2}} \cdot 3 \sqrt{6}+\color{blue}{ 5 \sqrt{2}} \cdot2 = \\ = 30 \sqrt{3} + 10 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{6}-2\right) } \cdot \left( 3 \sqrt{6} + 2\right) = \color{blue}{ 3 \sqrt{6}} \cdot 3 \sqrt{6}+\color{blue}{ 3 \sqrt{6}} \cdot2\color{blue}{-2} \cdot 3 \sqrt{6}\color{blue}{-2} \cdot2 = \\ = 54 + 6 \sqrt{6}- 6 \sqrt{6}-4 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 10. |
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