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