Answer
$$\frac{3\sqrt{2}}{2\sqrt{5}+5\sqrt{6}}=\frac{-6\sqrt{10}+30\sqrt{3}}{130}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{2}}{2\sqrt{5}+5\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{2}}{2\sqrt{5}+5\sqrt{6}}\frac{2\sqrt{5}-5\sqrt{6}}{2\sqrt{5}-5\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{10}-30\sqrt{3}}{20-10\sqrt{30}+10\sqrt{30}-150} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6\sqrt{10}-30\sqrt{3}}{-130} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-6\sqrt{10}+30\sqrt{3}}{130}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5}- 5 \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 \sqrt{2} } \cdot \left( 2 \sqrt{5}- 5 \sqrt{6}\right) = \color{blue}{ 3 \sqrt{2}} \cdot 2 \sqrt{5}+\color{blue}{ 3 \sqrt{2}} \cdot- 5 \sqrt{6} = \\ = 6 \sqrt{10}- 30 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{5} + 5 \sqrt{6}\right) } \cdot \left( 2 \sqrt{5}- 5 \sqrt{6}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot- 5 \sqrt{6}+\color{blue}{ 5 \sqrt{6}} \cdot 2 \sqrt{5}+\color{blue}{ 5 \sqrt{6}} \cdot- 5 \sqrt{6} = \\ = 20- 10 \sqrt{30} + 10 \sqrt{30}-150 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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