Question
Answer
$$\dfrac{2}{3+\sqrt{21}}=\dfrac{-3+\sqrt{21}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{3+\sqrt{21}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{3+\sqrt{21}}\frac{3-\sqrt{21}}{3-\sqrt{21}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6-2\sqrt{21}}{9-3\sqrt{21}+3\sqrt{21}-21} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6-2\sqrt{21}}{-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3-\sqrt{21}}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-3+\sqrt{21}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- \sqrt{21}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( 3- \sqrt{21}\right) = \color{blue}{2} \cdot3+\color{blue}{2} \cdot- \sqrt{21} = \\ = 6- 2 \sqrt{21} $$ Simplify denominator. $$ \color{blue}{ \left( 3 + \sqrt{21}\right) } \cdot \left( 3- \sqrt{21}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{21}+\color{blue}{ \sqrt{21}} \cdot3+\color{blue}{ \sqrt{21}} \cdot- \sqrt{21} = \\ = 9- 3 \sqrt{21} + 3 \sqrt{21}-21 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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