Question
Answer
$$\dfrac{3\sqrt{6}+5\sqrt{2}}{4\sqrt{6}-3\sqrt{2}}=\dfrac{51+29\sqrt{3}}{39}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{6}+5\sqrt{2}}{4\sqrt{6}-3\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{6}+5\sqrt{2}}{4\sqrt{6}-3\sqrt{2}}\frac{4\sqrt{6}+3\sqrt{2}}{4\sqrt{6}+3\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{72+18\sqrt{3}+40\sqrt{3}+30}{96+24\sqrt{3}-24\sqrt{3}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{102+58\sqrt{3}}{78} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{51+29\sqrt{3}}{39}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{6} + 3 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 \sqrt{6} + 5 \sqrt{2}\right) } \cdot \left( 4 \sqrt{6} + 3 \sqrt{2}\right) = \color{blue}{ 3 \sqrt{6}} \cdot 4 \sqrt{6}+\color{blue}{ 3 \sqrt{6}} \cdot 3 \sqrt{2}+\color{blue}{ 5 \sqrt{2}} \cdot 4 \sqrt{6}+\color{blue}{ 5 \sqrt{2}} \cdot 3 \sqrt{2} = \\ = 72 + 18 \sqrt{3} + 40 \sqrt{3} + 30 $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{6}- 3 \sqrt{2}\right) } \cdot \left( 4 \sqrt{6} + 3 \sqrt{2}\right) = \color{blue}{ 4 \sqrt{6}} \cdot 4 \sqrt{6}+\color{blue}{ 4 \sqrt{6}} \cdot 3 \sqrt{2}\color{blue}{- 3 \sqrt{2}} \cdot 4 \sqrt{6}\color{blue}{- 3 \sqrt{2}} \cdot 3 \sqrt{2} = \\ = 96 + 24 \sqrt{3}- 24 \sqrt{3}-18 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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