Answer
$$\frac{\sqrt{5}}{9+\sqrt{2}}=\frac{9\sqrt{5}-\sqrt{10}}{79}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}}{9+\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}}{9+\sqrt{2}}\frac{9-\sqrt{2}}{9-\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9\sqrt{5}-\sqrt{10}}{81-9\sqrt{2}+9\sqrt{2}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9\sqrt{5}-\sqrt{10}}{79}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 9- \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{5} } \cdot \left( 9- \sqrt{2}\right) = \color{blue}{ \sqrt{5}} \cdot9+\color{blue}{ \sqrt{5}} \cdot- \sqrt{2} = \\ = 9 \sqrt{5}- \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 9 + \sqrt{2}\right) } \cdot \left( 9- \sqrt{2}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot- \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot9+\color{blue}{ \sqrt{2}} \cdot- \sqrt{2} = \\ = 81- 9 \sqrt{2} + 9 \sqrt{2}-2 $$ |
| ③ | Simplify numerator and denominator |
This page was created using
Rationalize Denominator Calculator