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