Answer
$$\frac{3}{5+2\sqrt{3}}=\frac{15-6\sqrt{3}}{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3}{5+2\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3}{5+2\sqrt{3}}\frac{5-2\sqrt{3}}{5-2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{15-6\sqrt{3}}{25-10\sqrt{3}+10\sqrt{3}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15-6\sqrt{3}}{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- 2 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 } \cdot \left( 5- 2 \sqrt{3}\right) = \color{blue}{3} \cdot5+\color{blue}{3} \cdot- 2 \sqrt{3} = \\ = 15- 6 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 5 + 2 \sqrt{3}\right) } \cdot \left( 5- 2 \sqrt{3}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot5+\color{blue}{ 2 \sqrt{3}} \cdot- 2 \sqrt{3} = \\ = 25- 10 \sqrt{3} + 10 \sqrt{3}-12 $$ |
| ③ | Simplify numerator and denominator |
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