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