Question
Answer
$$\dfrac{\sqrt{15}}{8-2\sqrt{6}}=\dfrac{4\sqrt{15}+3\sqrt{10}}{20}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{15}}{8-2\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{15}}{8-2\sqrt{6}}\frac{8+2\sqrt{6}}{8+2\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8\sqrt{15}+6\sqrt{10}}{64+16\sqrt{6}-16\sqrt{6}-24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8\sqrt{15}+6\sqrt{10}}{40} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4\sqrt{15}+3\sqrt{10}}{20}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 8 + 2 \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{15} } \cdot \left( 8 + 2 \sqrt{6}\right) = \color{blue}{ \sqrt{15}} \cdot8+\color{blue}{ \sqrt{15}} \cdot 2 \sqrt{6} = \\ = 8 \sqrt{15} + 6 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 8- 2 \sqrt{6}\right) } \cdot \left( 8 + 2 \sqrt{6}\right) = \color{blue}{8} \cdot8+\color{blue}{8} \cdot 2 \sqrt{6}\color{blue}{- 2 \sqrt{6}} \cdot8\color{blue}{- 2 \sqrt{6}} \cdot 2 \sqrt{6} = \\ = 64 + 16 \sqrt{6}- 16 \sqrt{6}-24 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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