Answer
$$\frac{3+\sqrt{8}}{2-2\sqrt{8}}=-\frac{11+8\sqrt{2}}{14}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3+\sqrt{8}}{2-2\sqrt{8}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3+\sqrt{8}}{2-2\sqrt{8}}\frac{2+2\sqrt{8}}{2+2\sqrt{8}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6+12\sqrt{2}+4\sqrt{2}+16}{4+8\sqrt{2}-8\sqrt{2}-32} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{22+16\sqrt{2}}{-28} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{11+8\sqrt{2}}{-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{11+8\sqrt{2}}{14}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + 2 \sqrt{8}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 + \sqrt{8}\right) } \cdot \left( 2 + 2 \sqrt{8}\right) = \color{blue}{3} \cdot2+\color{blue}{3} \cdot 2 \sqrt{8}+\color{blue}{ \sqrt{8}} \cdot2+\color{blue}{ \sqrt{8}} \cdot 2 \sqrt{8} = \\ = 6 + 12 \sqrt{2} + 4 \sqrt{2} + 16 $$ Simplify denominator. $$ \color{blue}{ \left( 2- 2 \sqrt{8}\right) } \cdot \left( 2 + 2 \sqrt{8}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot 2 \sqrt{8}\color{blue}{- 2 \sqrt{8}} \cdot2\color{blue}{- 2 \sqrt{8}} \cdot 2 \sqrt{8} = \\ = 4 + 8 \sqrt{2}- 8 \sqrt{2}-32 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Place a negative sign in front of a fraction. |
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