Question
Answer
$$\dfrac{(\sqrt{5}+\sqrt{10})^2}{(\sqrt{5}-\sqrt{10})(\sqrt{5}+\sqrt{10})}=-3-2\sqrt{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(\sqrt{5}+\sqrt{10})^2}{(\sqrt{5}-\sqrt{10})(\sqrt{5}+\sqrt{10})}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5+5\sqrt{2}+5\sqrt{2}+10}{5+5\sqrt{2}-5\sqrt{2}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15+10\sqrt{2}}{-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3+2\sqrt{2}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{3+2\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }-(3+2\sqrt{2}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-3-2\sqrt{2}\end{aligned} $$ | |
| ① | $$ (\sqrt{5}+\sqrt{10})^2 = \left( \sqrt{5} + \sqrt{10} \right) \cdot \left( \sqrt{5} + \sqrt{10} \right) = 5 + 5 \sqrt{2} + 5 \sqrt{2} + 10 $$ |
| ② | $$ \color{blue}{ \left( \sqrt{5}- \sqrt{10}\right) } \cdot \left( \sqrt{5} + \sqrt{10}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot \sqrt{10}\color{blue}{- \sqrt{10}} \cdot \sqrt{5}\color{blue}{- \sqrt{10}} \cdot \sqrt{10} = \\ = 5 + 5 \sqrt{2}- 5 \sqrt{2}-10 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
| ⑤ | Place a negative sign in front of a fraction. |
| ⑥ | Remove the parenthesis by changing the sign of each term within them. |
This page was created using
Rationalize Denominator Calculator