Answer
$$\frac{3\sqrt{2}-2\sqrt{3}}{2\sqrt{3}+3\sqrt{2}}=-2\sqrt{6}+5$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{2}-2\sqrt{3}}{2\sqrt{3}+3\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{2}-2\sqrt{3}}{2\sqrt{3}+3\sqrt{2}}\frac{2\sqrt{3}-3\sqrt{2}}{2\sqrt{3}-3\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{6}-18-12+6\sqrt{6}}{12-6\sqrt{6}+6\sqrt{6}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12\sqrt{6}-30}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2\sqrt{6}-5}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-2\sqrt{6}+5}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-2\sqrt{6}+5\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{3}- 3 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 \sqrt{2}- 2 \sqrt{3}\right) } \cdot \left( 2 \sqrt{3}- 3 \sqrt{2}\right) = \color{blue}{ 3 \sqrt{2}} \cdot 2 \sqrt{3}+\color{blue}{ 3 \sqrt{2}} \cdot- 3 \sqrt{2}\color{blue}{- 2 \sqrt{3}} \cdot 2 \sqrt{3}\color{blue}{- 2 \sqrt{3}} \cdot- 3 \sqrt{2} = \\ = 6 \sqrt{6}-18-12 + 6 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{3} + 3 \sqrt{2}\right) } \cdot \left( 2 \sqrt{3}- 3 \sqrt{2}\right) = \color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot- 3 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot 2 \sqrt{3}+\color{blue}{ 3 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 12- 6 \sqrt{6} + 6 \sqrt{6}-18 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 6. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
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