Tap the blue circles to see an explanation.
$$ \begin{aligned}\frac{(\sqrt{3}-\sqrt{2})^2}{(\sqrt{3}+\sqrt{2})^2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3-\sqrt{6}-\sqrt{6}+2}{3+\sqrt{6}+\sqrt{6}+2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5-2\sqrt{6}}{5+2\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5-2\sqrt{6}}{5+2\sqrt{6}}\frac{5-2\sqrt{6}}{5-2\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{25-10\sqrt{6}-10\sqrt{6}+24}{25-10\sqrt{6}+10\sqrt{6}-24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{49-20\sqrt{6}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}49-20\sqrt{6}\end{aligned} $$ | |
① | $$ (\sqrt{3}-\sqrt{2})^2 = \left( \sqrt{3}- \sqrt{2} \right) \cdot \left( \sqrt{3}- \sqrt{2} \right) = 3- \sqrt{6}- \sqrt{6} + 2 $$ |
② | $$ (\sqrt{3}+\sqrt{2})^2 = \left( \sqrt{3} + \sqrt{2} \right) \cdot \left( \sqrt{3} + \sqrt{2} \right) = 3 + \sqrt{6} + \sqrt{6} + 2 $$ |
③ | Simplify numerator and denominator |
④ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- 2 \sqrt{6}} $$. |
⑤ | Multiply in a numerator. $$ \color{blue}{ \left( 5- 2 \sqrt{6}\right) } \cdot \left( 5- 2 \sqrt{6}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- 2 \sqrt{6}\color{blue}{- 2 \sqrt{6}} \cdot5\color{blue}{- 2 \sqrt{6}} \cdot- 2 \sqrt{6} = \\ = 25- 10 \sqrt{6}- 10 \sqrt{6} + 24 $$ Simplify denominator. $$ \color{blue}{ \left( 5 + 2 \sqrt{6}\right) } \cdot \left( 5- 2 \sqrt{6}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- 2 \sqrt{6}+\color{blue}{ 2 \sqrt{6}} \cdot5+\color{blue}{ 2 \sqrt{6}} \cdot- 2 \sqrt{6} = \\ = 25- 10 \sqrt{6} + 10 \sqrt{6}-24 $$ |
⑥ | Simplify numerator and denominator |
⑦ | Remove 1 from denominator. |