Question
Answer
$$\dfrac{2\sqrt{3}+3\sqrt{6}}{3\sqrt{3}-2\sqrt{6}}=18+13\sqrt{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{3}+3\sqrt{6}}{3\sqrt{3}-2\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{3}+3\sqrt{6}}{3\sqrt{3}-2\sqrt{6}}\frac{3\sqrt{3}+2\sqrt{6}}{3\sqrt{3}+2\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{18+12\sqrt{2}+27\sqrt{2}+36}{27+18\sqrt{2}-18\sqrt{2}-24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{54+39\sqrt{2}}{3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{18+13\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}18+13\sqrt{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{3} + 2 \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 \sqrt{3} + 3 \sqrt{6}\right) } \cdot \left( 3 \sqrt{3} + 2 \sqrt{6}\right) = \color{blue}{ 2 \sqrt{3}} \cdot 3 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{6}+\color{blue}{ 3 \sqrt{6}} \cdot 3 \sqrt{3}+\color{blue}{ 3 \sqrt{6}} \cdot 2 \sqrt{6} = \\ = 18 + 12 \sqrt{2} + 27 \sqrt{2} + 36 $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{3}- 2 \sqrt{6}\right) } \cdot \left( 3 \sqrt{3} + 2 \sqrt{6}\right) = \color{blue}{ 3 \sqrt{3}} \cdot 3 \sqrt{3}+\color{blue}{ 3 \sqrt{3}} \cdot 2 \sqrt{6}\color{blue}{- 2 \sqrt{6}} \cdot 3 \sqrt{3}\color{blue}{- 2 \sqrt{6}} \cdot 2 \sqrt{6} = \\ = 27 + 18 \sqrt{2}- 18 \sqrt{2}-24 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
| ⑤ | Remove 1 from denominator. |
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