Answer
$$\frac{6-4\sqrt{3}}{6+\sqrt{3}}=\frac{48-30\sqrt{3}}{33}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6-4\sqrt{3}}{6+\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6-4\sqrt{3}}{6+\sqrt{3}}\frac{6-\sqrt{3}}{6-\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{36-6\sqrt{3}-24\sqrt{3}+12}{36-6\sqrt{3}+6\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{48-30\sqrt{3}}{33}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6- \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 6- 4 \sqrt{3}\right) } \cdot \left( 6- \sqrt{3}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- \sqrt{3}\color{blue}{- 4 \sqrt{3}} \cdot6\color{blue}{- 4 \sqrt{3}} \cdot- \sqrt{3} = \\ = 36- 6 \sqrt{3}- 24 \sqrt{3} + 12 $$ Simplify denominator. $$ \color{blue}{ \left( 6 + \sqrt{3}\right) } \cdot \left( 6- \sqrt{3}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot6+\color{blue}{ \sqrt{3}} \cdot- \sqrt{3} = \\ = 36- 6 \sqrt{3} + 6 \sqrt{3}-3 $$ |
| ③ | Simplify numerator and denominator |
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