Answer
$$\frac{5}{\sqrt{75}-\sqrt{50}}=\sqrt{3}+\sqrt{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5}{\sqrt{75}-\sqrt{50}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5}{\sqrt{75}-\sqrt{50}}\frac{\sqrt{75}+\sqrt{50}}{\sqrt{75}+\sqrt{50}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{25\sqrt{3}+25\sqrt{2}}{75+25\sqrt{6}-25\sqrt{6}-50} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{25\sqrt{3}+25\sqrt{2}}{25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{3}+\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\sqrt{3}+\sqrt{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{75} + \sqrt{50}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 } \cdot \left( \sqrt{75} + \sqrt{50}\right) = \color{blue}{5} \cdot \sqrt{75}+\color{blue}{5} \cdot \sqrt{50} = \\ = 25 \sqrt{3} + 25 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{75}- \sqrt{50}\right) } \cdot \left( \sqrt{75} + \sqrt{50}\right) = \color{blue}{ \sqrt{75}} \cdot \sqrt{75}+\color{blue}{ \sqrt{75}} \cdot \sqrt{50}\color{blue}{- \sqrt{50}} \cdot \sqrt{75}\color{blue}{- \sqrt{50}} \cdot \sqrt{50} = \\ = 75 + 25 \sqrt{6}- 25 \sqrt{6}-50 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 25. |
| ⑤ | Remove 1 from denominator. |
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