Question
Answer
$$\dfrac{5\sqrt{2}-3\sqrt{5}}{\sqrt{18}+\sqrt{6}}=\dfrac{30-10\sqrt{3}-9\sqrt{10}+3\sqrt{30}}{12}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{2}-3\sqrt{5}}{\sqrt{18}+\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{2}-3\sqrt{5}}{\sqrt{18}+\sqrt{6}}\frac{\sqrt{18}-\sqrt{6}}{\sqrt{18}-\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30-10\sqrt{3}-9\sqrt{10}+3\sqrt{30}}{18-6\sqrt{3}+6\sqrt{3}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{30-10\sqrt{3}-9\sqrt{10}+3\sqrt{30}}{12}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{18}- \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 \sqrt{2}- 3 \sqrt{5}\right) } \cdot \left( \sqrt{18}- \sqrt{6}\right) = \color{blue}{ 5 \sqrt{2}} \cdot \sqrt{18}+\color{blue}{ 5 \sqrt{2}} \cdot- \sqrt{6}\color{blue}{- 3 \sqrt{5}} \cdot \sqrt{18}\color{blue}{- 3 \sqrt{5}} \cdot- \sqrt{6} = \\ = 30- 10 \sqrt{3}- 9 \sqrt{10} + 3 \sqrt{30} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{18} + \sqrt{6}\right) } \cdot \left( \sqrt{18}- \sqrt{6}\right) = \color{blue}{ \sqrt{18}} \cdot \sqrt{18}+\color{blue}{ \sqrt{18}} \cdot- \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot \sqrt{18}+\color{blue}{ \sqrt{6}} \cdot- \sqrt{6} = \\ = 18- 6 \sqrt{3} + 6 \sqrt{3}-6 $$ |
| ③ | Simplify numerator and denominator |
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