Question
Answer
$$\dfrac{15-2\sqrt{3}}{2\sqrt{3}+15}=\dfrac{-20\sqrt{3}+79}{71}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{15-2\sqrt{3}}{2\sqrt{3}+15}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{15-2\sqrt{3}}{2\sqrt{3}+15}\frac{2\sqrt{3}-15}{2\sqrt{3}-15} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30\sqrt{3}-225-12+30\sqrt{3}}{12-30\sqrt{3}+30\sqrt{3}-225} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{60\sqrt{3}-237}{-213} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{20\sqrt{3}-79}{-71} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-20\sqrt{3}+79}{71}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{3}-15} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 15- 2 \sqrt{3}\right) } \cdot \left( 2 \sqrt{3}-15\right) = \color{blue}{15} \cdot 2 \sqrt{3}+\color{blue}{15} \cdot-15\color{blue}{- 2 \sqrt{3}} \cdot 2 \sqrt{3}\color{blue}{- 2 \sqrt{3}} \cdot-15 = \\ = 30 \sqrt{3}-225-12 + 30 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{3} + 15\right) } \cdot \left( 2 \sqrt{3}-15\right) = \color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot-15+\color{blue}{15} \cdot 2 \sqrt{3}+\color{blue}{15} \cdot-15 = \\ = 12- 30 \sqrt{3} + 30 \sqrt{3}-225 $$ |
| 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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