Question
Answer
$$\dfrac{10}{10+\sqrt{10}}=\dfrac{10-\sqrt{10}}{9}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{10}{10+\sqrt{10}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{10}{10+\sqrt{10}}\frac{10-\sqrt{10}}{10-\sqrt{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{100-10\sqrt{10}}{100-10\sqrt{10}+10\sqrt{10}-10} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{100-10\sqrt{10}}{90} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{10-\sqrt{10}}{9}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 10- \sqrt{10}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 10 } \cdot \left( 10- \sqrt{10}\right) = \color{blue}{10} \cdot10+\color{blue}{10} \cdot- \sqrt{10} = \\ = 100- 10 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 10 + \sqrt{10}\right) } \cdot \left( 10- \sqrt{10}\right) = \color{blue}{10} \cdot10+\color{blue}{10} \cdot- \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot10+\color{blue}{ \sqrt{10}} \cdot- \sqrt{10} = \\ = 100- 10 \sqrt{10} + 10 \sqrt{10}-10 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 10. |
This page was created using
Rationalize Denominator Calculator