Answer
$$\frac{3\sqrt{3}+\sqrt{2}}{2\sqrt{5}+3\sqrt{2}}=\frac{6\sqrt{15}-9\sqrt{6}+2\sqrt{10}-6}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{3}+\sqrt{2}}{2\sqrt{5}+3\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{3}+\sqrt{2}}{2\sqrt{5}+3\sqrt{2}}\frac{2\sqrt{5}-3\sqrt{2}}{2\sqrt{5}-3\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{15}-9\sqrt{6}+2\sqrt{10}-6}{20-6\sqrt{10}+6\sqrt{10}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6\sqrt{15}-9\sqrt{6}+2\sqrt{10}-6}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5}- 3 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 \sqrt{3} + \sqrt{2}\right) } \cdot \left( 2 \sqrt{5}- 3 \sqrt{2}\right) = \color{blue}{ 3 \sqrt{3}} \cdot 2 \sqrt{5}+\color{blue}{ 3 \sqrt{3}} \cdot- 3 \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot 2 \sqrt{5}+\color{blue}{ \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 6 \sqrt{15}- 9 \sqrt{6} + 2 \sqrt{10}-6 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{5} + 3 \sqrt{2}\right) } \cdot \left( 2 \sqrt{5}- 3 \sqrt{2}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot- 3 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot 2 \sqrt{5}+\color{blue}{ 3 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 20- 6 \sqrt{10} + 6 \sqrt{10}-18 $$ |
| ③ | Simplify numerator and denominator |
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