Question
Answer
$$\dfrac{1+\sqrt{2}}{3-22\sqrt{2}}=-\dfrac{47+25\sqrt{2}}{959}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1+\sqrt{2}}{3-22\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1+\sqrt{2}}{3-22\sqrt{2}}\frac{3+22\sqrt{2}}{3+22\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3+22\sqrt{2}+3\sqrt{2}+44}{9+66\sqrt{2}-66\sqrt{2}-968} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{47+25\sqrt{2}}{-959} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{47+25\sqrt{2}}{959}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 + 22 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 1 + \sqrt{2}\right) } \cdot \left( 3 + 22 \sqrt{2}\right) = \color{blue}{1} \cdot3+\color{blue}{1} \cdot 22 \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot3+\color{blue}{ \sqrt{2}} \cdot 22 \sqrt{2} = \\ = 3 + 22 \sqrt{2} + 3 \sqrt{2} + 44 $$ Simplify denominator. $$ \color{blue}{ \left( 3- 22 \sqrt{2}\right) } \cdot \left( 3 + 22 \sqrt{2}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot 22 \sqrt{2}\color{blue}{- 22 \sqrt{2}} \cdot3\color{blue}{- 22 \sqrt{2}} \cdot 22 \sqrt{2} = \\ = 9 + 66 \sqrt{2}- 66 \sqrt{2}-968 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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