Question
Answer
$$\dfrac{9+\sqrt{8}}{4-\sqrt{18}}=-\dfrac{48+35\sqrt{2}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{9+\sqrt{8}}{4-\sqrt{18}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{9+\sqrt{8}}{4-\sqrt{18}}\frac{4+\sqrt{18}}{4+\sqrt{18}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{36+27\sqrt{2}+8\sqrt{2}+12}{16+12\sqrt{2}-12\sqrt{2}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{48+35\sqrt{2}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{48+35\sqrt{2}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 + \sqrt{18}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 9 + \sqrt{8}\right) } \cdot \left( 4 + \sqrt{18}\right) = \color{blue}{9} \cdot4+\color{blue}{9} \cdot \sqrt{18}+\color{blue}{ \sqrt{8}} \cdot4+\color{blue}{ \sqrt{8}} \cdot \sqrt{18} = \\ = 36 + 27 \sqrt{2} + 8 \sqrt{2} + 12 $$ Simplify denominator. $$ \color{blue}{ \left( 4- \sqrt{18}\right) } \cdot \left( 4 + \sqrt{18}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot \sqrt{18}\color{blue}{- \sqrt{18}} \cdot4\color{blue}{- \sqrt{18}} \cdot \sqrt{18} = \\ = 16 + 12 \sqrt{2}- 12 \sqrt{2}-18 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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