Question
Answer
$$\dfrac{\sqrt{2}-\sqrt{10}}{\sqrt{15}+\sqrt{12}}=\dfrac{3\sqrt{30}-7\sqrt{6}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}-\sqrt{10}}{\sqrt{15}+\sqrt{12}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}-\sqrt{10}}{\sqrt{15}+\sqrt{12}}\frac{\sqrt{15}-\sqrt{12}}{\sqrt{15}-\sqrt{12}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{30}-2\sqrt{6}-5\sqrt{6}+2\sqrt{30}}{15-6\sqrt{5}+6\sqrt{5}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{3\sqrt{30}-7\sqrt{6}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{15}- \sqrt{12}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{2}- \sqrt{10}\right) } \cdot \left( \sqrt{15}- \sqrt{12}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{15}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{12}\color{blue}{- \sqrt{10}} \cdot \sqrt{15}\color{blue}{- \sqrt{10}} \cdot- \sqrt{12} = \\ = \sqrt{30}- 2 \sqrt{6}- 5 \sqrt{6} + 2 \sqrt{30} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{15} + \sqrt{12}\right) } \cdot \left( \sqrt{15}- \sqrt{12}\right) = \color{blue}{ \sqrt{15}} \cdot \sqrt{15}+\color{blue}{ \sqrt{15}} \cdot- \sqrt{12}+\color{blue}{ \sqrt{12}} \cdot \sqrt{15}+\color{blue}{ \sqrt{12}} \cdot- \sqrt{12} = \\ = 15- 6 \sqrt{5} + 6 \sqrt{5}-12 $$ |
| ③ | Simplify numerator and denominator |
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