Question
Answer
$$\dfrac{\sqrt{48}+\sqrt{18}}{\sqrt{48}-\sqrt{18}}=\dfrac{11+4\sqrt{6}}{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{48}+\sqrt{18}}{\sqrt{48}-\sqrt{18}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{48}+\sqrt{18}}{\sqrt{48}-\sqrt{18}}\frac{\sqrt{48}+\sqrt{18}}{\sqrt{48}+\sqrt{18}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{48+12\sqrt{6}+12\sqrt{6}+18}{48+12\sqrt{6}-12\sqrt{6}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{66+24\sqrt{6}}{30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{11+4\sqrt{6}}{5}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{48} + \sqrt{18}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{48} + \sqrt{18}\right) } \cdot \left( \sqrt{48} + \sqrt{18}\right) = \color{blue}{ \sqrt{48}} \cdot \sqrt{48}+\color{blue}{ \sqrt{48}} \cdot \sqrt{18}+\color{blue}{ \sqrt{18}} \cdot \sqrt{48}+\color{blue}{ \sqrt{18}} \cdot \sqrt{18} = \\ = 48 + 12 \sqrt{6} + 12 \sqrt{6} + 18 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{48}- \sqrt{18}\right) } \cdot \left( \sqrt{48} + \sqrt{18}\right) = \color{blue}{ \sqrt{48}} \cdot \sqrt{48}+\color{blue}{ \sqrt{48}} \cdot \sqrt{18}\color{blue}{- \sqrt{18}} \cdot \sqrt{48}\color{blue}{- \sqrt{18}} \cdot \sqrt{18} = \\ = 48 + 12 \sqrt{6}- 12 \sqrt{6}-18 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 6. |
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