Answer
$$\frac{2\sqrt{5}+3\sqrt{2}}{4\sqrt{5}-5\sqrt{2}}=\frac{35+11\sqrt{10}}{15}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{5}+3\sqrt{2}}{4\sqrt{5}-5\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{5}+3\sqrt{2}}{4\sqrt{5}-5\sqrt{2}}\frac{4\sqrt{5}+5\sqrt{2}}{4\sqrt{5}+5\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{40+10\sqrt{10}+12\sqrt{10}+30}{80+20\sqrt{10}-20\sqrt{10}-50} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{70+22\sqrt{10}}{30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{35+11\sqrt{10}}{15}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{5} + 5 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 \sqrt{5} + 3 \sqrt{2}\right) } \cdot \left( 4 \sqrt{5} + 5 \sqrt{2}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 4 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot 5 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot 4 \sqrt{5}+\color{blue}{ 3 \sqrt{2}} \cdot 5 \sqrt{2} = \\ = 40 + 10 \sqrt{10} + 12 \sqrt{10} + 30 $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{5}- 5 \sqrt{2}\right) } \cdot \left( 4 \sqrt{5} + 5 \sqrt{2}\right) = \color{blue}{ 4 \sqrt{5}} \cdot 4 \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot 5 \sqrt{2}\color{blue}{- 5 \sqrt{2}} \cdot 4 \sqrt{5}\color{blue}{- 5 \sqrt{2}} \cdot 5 \sqrt{2} = \\ = 80 + 20 \sqrt{10}- 20 \sqrt{10}-50 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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