Question
Answer
$$\dfrac{1}{6+2\sqrt{15}}=\dfrac{-3+\sqrt{15}}{12}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{6+2\sqrt{15}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{6+2\sqrt{15}}\frac{6-2\sqrt{15}}{6-2\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6-2\sqrt{15}}{36-12\sqrt{15}+12\sqrt{15}-60} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6-2\sqrt{15}}{-24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3-\sqrt{15}}{-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-3+\sqrt{15}}{12}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6- 2 \sqrt{15}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 6- 2 \sqrt{15}\right) = \color{blue}{1} \cdot6+\color{blue}{1} \cdot- 2 \sqrt{15} = \\ = 6- 2 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 6 + 2 \sqrt{15}\right) } \cdot \left( 6- 2 \sqrt{15}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- 2 \sqrt{15}+\color{blue}{ 2 \sqrt{15}} \cdot6+\color{blue}{ 2 \sqrt{15}} \cdot- 2 \sqrt{15} = \\ = 36- 12 \sqrt{15} + 12 \sqrt{15}-60 $$ |
| 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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