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