Answer
$$\frac{2+\sqrt{2}}{3-\sqrt{3}}=\frac{6+2\sqrt{3}+3\sqrt{2}+\sqrt{6}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2+\sqrt{2}}{3-\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2+\sqrt{2}}{3-\sqrt{3}}\frac{3+\sqrt{3}}{3+\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6+2\sqrt{3}+3\sqrt{2}+\sqrt{6}}{9+3\sqrt{3}-3\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6+2\sqrt{3}+3\sqrt{2}+\sqrt{6}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 + \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 + \sqrt{2}\right) } \cdot \left( 3 + \sqrt{3}\right) = \color{blue}{2} \cdot3+\color{blue}{2} \cdot \sqrt{3}+\color{blue}{ \sqrt{2}} \cdot3+\color{blue}{ \sqrt{2}} \cdot \sqrt{3} = \\ = 6 + 2 \sqrt{3} + 3 \sqrt{2} + \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 3- \sqrt{3}\right) } \cdot \left( 3 + \sqrt{3}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot \sqrt{3}\color{blue}{- \sqrt{3}} \cdot3\color{blue}{- \sqrt{3}} \cdot \sqrt{3} = \\ = 9 + 3 \sqrt{3}- 3 \sqrt{3}-3 $$ |
| ③ | Simplify numerator and denominator |
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