Question
Answer
$$\dfrac{-\sqrt{15}}{4+8\sqrt{3}}=\dfrac{\sqrt{15}-6\sqrt{5}}{44}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{-\sqrt{15}}{4+8\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-\sqrt{15}}{4+8\sqrt{3}}\frac{4-8\sqrt{3}}{4-8\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-4\sqrt{15}+24\sqrt{5}}{16-32\sqrt{3}+32\sqrt{3}-192} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-4\sqrt{15}+24\sqrt{5}}{-176} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-\sqrt{15}+6\sqrt{5}}{-44} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{\sqrt{15}-6\sqrt{5}}{44}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4- 8 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ - \sqrt{15} } \cdot \left( 4- 8 \sqrt{3}\right) = \color{blue}{- \sqrt{15}} \cdot4\color{blue}{- \sqrt{15}} \cdot- 8 \sqrt{3} = \\ = - 4 \sqrt{15} + 24 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 4 + 8 \sqrt{3}\right) } \cdot \left( 4- 8 \sqrt{3}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot- 8 \sqrt{3}+\color{blue}{ 8 \sqrt{3}} \cdot4+\color{blue}{ 8 \sqrt{3}} \cdot- 8 \sqrt{3} = \\ = 16- 32 \sqrt{3} + 32 \sqrt{3}-192 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
| ⑤ | Multiply both numerator and denominator by -1. |
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