Question
Answer
$$\dfrac{3}{-3+2\sqrt{6}}=\dfrac{3+2\sqrt{6}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3}{-3+2\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3}{-3+2\sqrt{6}}\frac{-3-2\sqrt{6}}{-3-2\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-9-6\sqrt{6}}{9+6\sqrt{6}-6\sqrt{6}-24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-9-6\sqrt{6}}{-15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-3-2\sqrt{6}}{-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{3+2\sqrt{6}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -3- 2 \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 } \cdot \left( -3- 2 \sqrt{6}\right) = \color{blue}{3} \cdot-3+\color{blue}{3} \cdot- 2 \sqrt{6} = \\ = -9- 6 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( -3 + 2 \sqrt{6}\right) } \cdot \left( -3- 2 \sqrt{6}\right) = \color{blue}{-3} \cdot-3\color{blue}{-3} \cdot- 2 \sqrt{6}+\color{blue}{ 2 \sqrt{6}} \cdot-3+\color{blue}{ 2 \sqrt{6}} \cdot- 2 \sqrt{6} = \\ = 9 + 6 \sqrt{6}- 6 \sqrt{6}-24 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
| ⑤ | Multiply both numerator and denominator by -1. |
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