Answer
$$\frac{2}{2+\sqrt{2}}-\frac{3}{3+\sqrt{3}}+\frac{6}{\sqrt{2}+\sqrt{3}}=\frac{1-14\sqrt{2}+13\sqrt{3}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{2+\sqrt{2}}-\frac{3}{3+\sqrt{3}}+\frac{6}{\sqrt{2}+\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4-2\sqrt{2}}{4-2\sqrt{2}+2\sqrt{2}-2}-\frac{9-3\sqrt{3}}{9-3\sqrt{3}+3\sqrt{3}-3}+\frac{6\sqrt{2}-6\sqrt{3}}{2-\sqrt{6}+\sqrt{6}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4-2\sqrt{2}}{2}-\frac{9-3\sqrt{3}}{6}+\frac{6\sqrt{2}-6\sqrt{3}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2-\sqrt{2}}{1}-\frac{3-\sqrt{3}}{2}+\frac{-6\sqrt{2}+6\sqrt{3}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2-\sqrt{2}-\frac{3-\sqrt{3}}{2}-6\sqrt{2}+6\sqrt{3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{4-2\sqrt{2}-3+\sqrt{3}}{2}-6\sqrt{2}+6\sqrt{3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{1-2\sqrt{2}+\sqrt{3}}{2}-6\sqrt{2}+6\sqrt{3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{1-2\sqrt{2}+\sqrt{3}-12\sqrt{2}+12\sqrt{3}}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{1-14\sqrt{2}+13\sqrt{3}}{2}\end{aligned} $$ | |
| ① | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( 2- \sqrt{2}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- \sqrt{2} = \\ = 4- 2 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 2 + \sqrt{2}\right) } \cdot \left( 2- \sqrt{2}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot2+\color{blue}{ \sqrt{2}} \cdot- \sqrt{2} = \\ = 4- 2 \sqrt{2} + 2 \sqrt{2}-2 $$Multiply in a numerator. $$ \color{blue}{ 3 } \cdot \left( 3- \sqrt{3}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{3} = \\ = 9- 3 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 3 + \sqrt{3}\right) } \cdot \left( 3- \sqrt{3}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot3+\color{blue}{ \sqrt{3}} \cdot- \sqrt{3} = \\ = 9- 3 \sqrt{3} + 3 \sqrt{3}-3 $$Multiply in a numerator. $$ \color{blue}{ 6 } \cdot \left( \sqrt{2}- \sqrt{3}\right) = \color{blue}{6} \cdot \sqrt{2}+\color{blue}{6} \cdot- \sqrt{3} = \\ = 6 \sqrt{2}- 6 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{2} + \sqrt{3}\right) } \cdot \left( \sqrt{2}- \sqrt{3}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot \sqrt{2}+\color{blue}{ \sqrt{3}} \cdot- \sqrt{3} = \\ = 2- \sqrt{6} + \sqrt{6}-3 $$ |
| ② | Simplify numerator and denominatorSimplify numerator and denominatorSimplify numerator and denominator |
| ③ | Divide both numerator and denominator by 2.Divide both numerator and denominator by 3.Multiply both numerator and denominator by -1. |
| ④ | Remove 1 from denominator.Remove 1 from denominator. |
| ⑤ | $$ 2-\sqrt{2}-\frac{3-\sqrt{3}}{2}
= 2-\sqrt{2} \cdot \color{blue}{\frac{ 2 }{ 2}} - \frac{3-\sqrt{3}}{2} \cdot \color{blue}{\frac{ 1 }{ 1}}
= \frac{4-2\sqrt{2}-3+\sqrt{3}}{2} $$ |
| ⑥ | Simplify numerator and denominator |
| ⑦ | $$ \frac{1-2\sqrt{2}+\sqrt{3}}{2}-6\sqrt{2}+6\sqrt{3}
= \frac{1-2\sqrt{2}+\sqrt{3}}{2} \cdot \color{blue}{\frac{ 1 }{ 1}} + -6\sqrt{2}+6\sqrt{3} \cdot \color{blue}{\frac{ 2 }{ 2}}
= \frac{1-2\sqrt{2}+\sqrt{3}-12\sqrt{2}+12\sqrt{3}}{2} $$ |
| ⑧ | Simplify numerator and denominator |
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