Question
Answer
$$\dfrac{3\sqrt{6}+4\sqrt{3}}{4\sqrt{6}+2\sqrt{3}}=\dfrac{8+5\sqrt{2}}{14}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{6}+4\sqrt{3}}{4\sqrt{6}+2\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{6}+4\sqrt{3}}{4\sqrt{6}+2\sqrt{3}}\frac{4\sqrt{6}-2\sqrt{3}}{4\sqrt{6}-2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{72-18\sqrt{2}+48\sqrt{2}-24}{96-24\sqrt{2}+24\sqrt{2}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{48+30\sqrt{2}}{84} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{8+5\sqrt{2}}{14}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{6}- 2 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 \sqrt{6} + 4 \sqrt{3}\right) } \cdot \left( 4 \sqrt{6}- 2 \sqrt{3}\right) = \color{blue}{ 3 \sqrt{6}} \cdot 4 \sqrt{6}+\color{blue}{ 3 \sqrt{6}} \cdot- 2 \sqrt{3}+\color{blue}{ 4 \sqrt{3}} \cdot 4 \sqrt{6}+\color{blue}{ 4 \sqrt{3}} \cdot- 2 \sqrt{3} = \\ = 72- 18 \sqrt{2} + 48 \sqrt{2}-24 $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{6} + 2 \sqrt{3}\right) } \cdot \left( 4 \sqrt{6}- 2 \sqrt{3}\right) = \color{blue}{ 4 \sqrt{6}} \cdot 4 \sqrt{6}+\color{blue}{ 4 \sqrt{6}} \cdot- 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot 4 \sqrt{6}+\color{blue}{ 2 \sqrt{3}} \cdot- 2 \sqrt{3} = \\ = 96- 24 \sqrt{2} + 24 \sqrt{2}-12 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 6. |
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