Answer
$$\frac{3-6\sqrt{3}}{18+\sqrt{3}}=\frac{24-37\sqrt{3}}{107}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3-6\sqrt{3}}{18+\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3-6\sqrt{3}}{18+\sqrt{3}}\frac{18-\sqrt{3}}{18-\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{54-3\sqrt{3}-108\sqrt{3}+18}{324-18\sqrt{3}+18\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{72-111\sqrt{3}}{321} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{24-37\sqrt{3}}{107}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 18- \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3- 6 \sqrt{3}\right) } \cdot \left( 18- \sqrt{3}\right) = \color{blue}{3} \cdot18+\color{blue}{3} \cdot- \sqrt{3}\color{blue}{- 6 \sqrt{3}} \cdot18\color{blue}{- 6 \sqrt{3}} \cdot- \sqrt{3} = \\ = 54- 3 \sqrt{3}- 108 \sqrt{3} + 18 $$ Simplify denominator. $$ \color{blue}{ \left( 18 + \sqrt{3}\right) } \cdot \left( 18- \sqrt{3}\right) = \color{blue}{18} \cdot18+\color{blue}{18} \cdot- \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot18+\color{blue}{ \sqrt{3}} \cdot- \sqrt{3} = \\ = 324- 18 \sqrt{3} + 18 \sqrt{3}-3 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
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