Question
Answer
$$\dfrac{2\sqrt{5}-3}{6+3\sqrt{3}}=\dfrac{4\sqrt{5}-2\sqrt{15}-6+3\sqrt{3}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{5}-3}{6+3\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{5}-3}{6+3\sqrt{3}}\frac{6-3\sqrt{3}}{6-3\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{12\sqrt{5}-6\sqrt{15}-18+9\sqrt{3}}{36-18\sqrt{3}+18\sqrt{3}-27} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12\sqrt{5}-6\sqrt{15}-18+9\sqrt{3}}{9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4\sqrt{5}-2\sqrt{15}-6+3\sqrt{3}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6- 3 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 \sqrt{5}-3\right) } \cdot \left( 6- 3 \sqrt{3}\right) = \color{blue}{ 2 \sqrt{5}} \cdot6+\color{blue}{ 2 \sqrt{5}} \cdot- 3 \sqrt{3}\color{blue}{-3} \cdot6\color{blue}{-3} \cdot- 3 \sqrt{3} = \\ = 12 \sqrt{5}- 6 \sqrt{15}-18 + 9 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 6 + 3 \sqrt{3}\right) } \cdot \left( 6- 3 \sqrt{3}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- 3 \sqrt{3}+\color{blue}{ 3 \sqrt{3}} \cdot6+\color{blue}{ 3 \sqrt{3}} \cdot- 3 \sqrt{3} = \\ = 36- 18 \sqrt{3} + 18 \sqrt{3}-27 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
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