Question
Answer
$$\dfrac{\sqrt{8}+\sqrt{15}}{4\sqrt{18}-\sqrt{3}}=\dfrac{48+2\sqrt{6}+12\sqrt{30}+3\sqrt{5}}{285}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{8}+\sqrt{15}}{4\sqrt{18}-\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{8}+\sqrt{15}}{4\sqrt{18}-\sqrt{3}}\frac{4\sqrt{18}+\sqrt{3}}{4\sqrt{18}+\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{48+2\sqrt{6}+12\sqrt{30}+3\sqrt{5}}{288+12\sqrt{6}-12\sqrt{6}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{48+2\sqrt{6}+12\sqrt{30}+3\sqrt{5}}{285}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{18} + \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{8} + \sqrt{15}\right) } \cdot \left( 4 \sqrt{18} + \sqrt{3}\right) = \color{blue}{ \sqrt{8}} \cdot 4 \sqrt{18}+\color{blue}{ \sqrt{8}} \cdot \sqrt{3}+\color{blue}{ \sqrt{15}} \cdot 4 \sqrt{18}+\color{blue}{ \sqrt{15}} \cdot \sqrt{3} = \\ = 48 + 2 \sqrt{6} + 12 \sqrt{30} + 3 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{18}- \sqrt{3}\right) } \cdot \left( 4 \sqrt{18} + \sqrt{3}\right) = \color{blue}{ 4 \sqrt{18}} \cdot 4 \sqrt{18}+\color{blue}{ 4 \sqrt{18}} \cdot \sqrt{3}\color{blue}{- \sqrt{3}} \cdot 4 \sqrt{18}\color{blue}{- \sqrt{3}} \cdot \sqrt{3} = \\ = 288 + 12 \sqrt{6}- 12 \sqrt{6}-3 $$ |
| ③ | Simplify numerator and denominator |
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