Answer
$$\frac{4\sqrt{5}-7\sqrt{2}}{2\sqrt{5}+\sqrt{2}}=3-\sqrt{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4\sqrt{5}-7\sqrt{2}}{2\sqrt{5}+\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4\sqrt{5}-7\sqrt{2}}{2\sqrt{5}+\sqrt{2}}\frac{2\sqrt{5}-\sqrt{2}}{2\sqrt{5}-\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{40-4\sqrt{10}-14\sqrt{10}+14}{20-2\sqrt{10}+2\sqrt{10}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{54-18\sqrt{10}}{18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3-\sqrt{10}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}3-\sqrt{10}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5}- \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4 \sqrt{5}- 7 \sqrt{2}\right) } \cdot \left( 2 \sqrt{5}- \sqrt{2}\right) = \color{blue}{ 4 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot- \sqrt{2}\color{blue}{- 7 \sqrt{2}} \cdot 2 \sqrt{5}\color{blue}{- 7 \sqrt{2}} \cdot- \sqrt{2} = \\ = 40- 4 \sqrt{10}- 14 \sqrt{10} + 14 $$ 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 18. |
| ⑤ | Remove 1 from denominator. |
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