Question
Answer
$$\dfrac{\sqrt{2}}{2+3\sqrt{2}}=\dfrac{-\sqrt{2}+3}{7}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}}{2+3\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}}{2+3\sqrt{2}}\frac{2-3\sqrt{2}}{2-3\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{2}-6}{4-6\sqrt{2}+6\sqrt{2}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{2}-6}{-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{2}-3}{-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-\sqrt{2}+3}{7}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2- 3 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{2} } \cdot \left( 2- 3 \sqrt{2}\right) = \color{blue}{ \sqrt{2}} \cdot2+\color{blue}{ \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 2 \sqrt{2}-6 $$ Simplify denominator. $$ \color{blue}{ \left( 2 + 3 \sqrt{2}\right) } \cdot \left( 2- 3 \sqrt{2}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- 3 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot2+\color{blue}{ 3 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 4- 6 \sqrt{2} + 6 \sqrt{2}-18 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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