Question
Answer
$$\dfrac{1}{\sqrt{5}+\sqrt{2}+\sqrt{5}}=\dfrac{2\sqrt{5}-\sqrt{2}}{18}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{\sqrt{5}+\sqrt{2}+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{2\sqrt{5}+\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1}{2\sqrt{5}+\sqrt{2}}\frac{2\sqrt{5}-\sqrt{2}}{2\sqrt{5}-\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{5}-\sqrt{2}}{20-2\sqrt{10}+2\sqrt{10}-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2\sqrt{5}-\sqrt{2}}{18}\end{aligned} $$ | |
| ① | Simplify numerator and denominator |
| ② | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5}- \sqrt{2}} $$. |
| ③ | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 2 \sqrt{5}- \sqrt{2}\right) = \color{blue}{1} \cdot 2 \sqrt{5}+\color{blue}{1} \cdot- \sqrt{2} = \\ = 2 \sqrt{5}- \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{5} + \sqrt{2}\right) } \cdot \left( 2 \sqrt{5}- \sqrt{2}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot- \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot 2 \sqrt{5}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{2} = \\ = 20- 2 \sqrt{10} + 2 \sqrt{10}-2 $$ |
| ④ | Simplify numerator and denominator |
This page was created using
Rationalize Denominator Calculator