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