Answer
$$\frac{1}{4-\sqrt{8}}=\frac{2+\sqrt{2}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{4-\sqrt{8}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{4-\sqrt{8}}\frac{4+\sqrt{8}}{4+\sqrt{8}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4+2\sqrt{2}}{16+8\sqrt{2}-8\sqrt{2}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4+2\sqrt{2}}{8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2+\sqrt{2}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 + \sqrt{8}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 4 + \sqrt{8}\right) = \color{blue}{1} \cdot4+\color{blue}{1} \cdot \sqrt{8} = \\ = 4 + 2 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 4- \sqrt{8}\right) } \cdot \left( 4 + \sqrt{8}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot \sqrt{8}\color{blue}{- \sqrt{8}} \cdot4\color{blue}{- \sqrt{8}} \cdot \sqrt{8} = \\ = 16 + 8 \sqrt{2}- 8 \sqrt{2}-8 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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