Question
Answer
$$\dfrac{9}{-9+\sqrt{2}}=\dfrac{-81-9\sqrt{2}}{79}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{9}{-9+\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{9}{-9+\sqrt{2}}\frac{-9-\sqrt{2}}{-9-\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-81-9\sqrt{2}}{81+9\sqrt{2}-9\sqrt{2}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-81-9\sqrt{2}}{79}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -9- \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 9 } \cdot \left( -9- \sqrt{2}\right) = \color{blue}{9} \cdot-9+\color{blue}{9} \cdot- \sqrt{2} = \\ = -81- 9 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( -9 + \sqrt{2}\right) } \cdot \left( -9- \sqrt{2}\right) = \color{blue}{-9} \cdot-9\color{blue}{-9} \cdot- \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot-9+\color{blue}{ \sqrt{2}} \cdot- \sqrt{2} = \\ = 81 + 9 \sqrt{2}- 9 \sqrt{2}-2 $$ |
| ③ | Simplify numerator and denominator |
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