Answer
$$\frac{10\sqrt{2}+25}{5+2\sqrt{2}}=5$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{10\sqrt{2}+25}{5+2\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{10\sqrt{2}+25}{5+2\sqrt{2}}\frac{5-2\sqrt{2}}{5-2\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{50\sqrt{2}-40+125-50\sqrt{2}}{25-10\sqrt{2}+10\sqrt{2}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{85}{17} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}} \frac{ 85 : \color{orangered}{ 17 } }{ 17 : \color{orangered}{ 17 }} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{5}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}5\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5- 2 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 10 \sqrt{2} + 25\right) } \cdot \left( 5- 2 \sqrt{2}\right) = \color{blue}{ 10 \sqrt{2}} \cdot5+\color{blue}{ 10 \sqrt{2}} \cdot- 2 \sqrt{2}+\color{blue}{25} \cdot5+\color{blue}{25} \cdot- 2 \sqrt{2} = \\ = 50 \sqrt{2}-40 + 125- 50 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 5 + 2 \sqrt{2}\right) } \cdot \left( 5- 2 \sqrt{2}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot- 2 \sqrt{2}+\color{blue}{ 2 \sqrt{2}} \cdot5+\color{blue}{ 2 \sqrt{2}} \cdot- 2 \sqrt{2} = \\ = 25- 10 \sqrt{2} + 10 \sqrt{2}-8 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both the top and bottom numbers by $ \color{orangered}{ 17 } $. |
| ⑤ | Remove 1 from denominator. |
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