Answer
$$\frac{\sqrt{4}+\sqrt{10}}{5\sqrt{12}-\sqrt{5}}=\frac{20\sqrt{3}+2\sqrt{5}+10\sqrt{30}+5\sqrt{2}}{295}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{4}+\sqrt{10}}{5\sqrt{12}-\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{4}+\sqrt{10}}{5\sqrt{12}-\sqrt{5}}\frac{5\sqrt{12}+\sqrt{5}}{5\sqrt{12}+\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20\sqrt{3}+2\sqrt{5}+10\sqrt{30}+5\sqrt{2}}{300+10\sqrt{15}-10\sqrt{15}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{20\sqrt{3}+2\sqrt{5}+10\sqrt{30}+5\sqrt{2}}{295}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{12} + \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{4} + \sqrt{10}\right) } \cdot \left( 5 \sqrt{12} + \sqrt{5}\right) = \color{blue}{ \sqrt{4}} \cdot 5 \sqrt{12}+\color{blue}{ \sqrt{4}} \cdot \sqrt{5}+\color{blue}{ \sqrt{10}} \cdot 5 \sqrt{12}+\color{blue}{ \sqrt{10}} \cdot \sqrt{5} = \\ = 20 \sqrt{3} + 2 \sqrt{5} + 10 \sqrt{30} + 5 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{12}- \sqrt{5}\right) } \cdot \left( 5 \sqrt{12} + \sqrt{5}\right) = \color{blue}{ 5 \sqrt{12}} \cdot 5 \sqrt{12}+\color{blue}{ 5 \sqrt{12}} \cdot \sqrt{5}\color{blue}{- \sqrt{5}} \cdot 5 \sqrt{12}\color{blue}{- \sqrt{5}} \cdot \sqrt{5} = \\ = 300 + 10 \sqrt{15}- 10 \sqrt{15}-5 $$ |
| ③ | Simplify numerator and denominator |
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