Question
Answer
$$\dfrac{2}{2\sqrt{3}+5}=\dfrac{-4\sqrt{3}+10}{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{2\sqrt{3}+5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{2\sqrt{3}+5}\frac{2\sqrt{3}-5}{2\sqrt{3}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4\sqrt{3}-10}{12-10\sqrt{3}+10\sqrt{3}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4\sqrt{3}-10}{-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-4\sqrt{3}+10}{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{3}-5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( 2 \sqrt{3}-5\right) = \color{blue}{2} \cdot 2 \sqrt{3}+\color{blue}{2} \cdot-5 = \\ = 4 \sqrt{3}-10 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{3} + 5\right) } \cdot \left( 2 \sqrt{3}-5\right) = \color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot-5+\color{blue}{5} \cdot 2 \sqrt{3}+\color{blue}{5} \cdot-5 = \\ = 12- 10 \sqrt{3} + 10 \sqrt{3}-25 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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