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