Answer
$$\frac{\sqrt{5}-\sqrt{2}}{2\sqrt{5}+\sqrt{2}}=\frac{4-\sqrt{10}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}-\sqrt{2}}{2\sqrt{5}+\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}-\sqrt{2}}{2\sqrt{5}+\sqrt{2}}\frac{2\sqrt{5}-\sqrt{2}}{2\sqrt{5}-\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10-\sqrt{10}-2\sqrt{10}+2}{20-2\sqrt{10}+2\sqrt{10}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12-3\sqrt{10}}{18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4-\sqrt{10}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5}- \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{5}- \sqrt{2}\right) } \cdot \left( 2 \sqrt{5}- \sqrt{2}\right) = \color{blue}{ \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{2}\color{blue}{- \sqrt{2}} \cdot 2 \sqrt{5}\color{blue}{- \sqrt{2}} \cdot- \sqrt{2} = \\ = 10- \sqrt{10}- 2 \sqrt{10} + 2 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{5} + \sqrt{2}\right) } \cdot \left( 2 \sqrt{5}- \sqrt{2}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot- \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot 2 \sqrt{5}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{2} = \\ = 20- 2 \sqrt{10} + 2 \sqrt{10}-2 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
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