Answer
$$\frac{3}{2+5\sqrt{5}}=\frac{-6+15\sqrt{5}}{121}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3}{2+5\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3}{2+5\sqrt{5}}\frac{2-5\sqrt{5}}{2-5\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6-15\sqrt{5}}{4-10\sqrt{5}+10\sqrt{5}-125} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6-15\sqrt{5}}{-121} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-6+15\sqrt{5}}{121}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2- 5 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 } \cdot \left( 2- 5 \sqrt{5}\right) = \color{blue}{3} \cdot2+\color{blue}{3} \cdot- 5 \sqrt{5} = \\ = 6- 15 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 2 + 5 \sqrt{5}\right) } \cdot \left( 2- 5 \sqrt{5}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot- 5 \sqrt{5}+\color{blue}{ 5 \sqrt{5}} \cdot2+\color{blue}{ 5 \sqrt{5}} \cdot- 5 \sqrt{5} = \\ = 4- 10 \sqrt{5} + 10 \sqrt{5}-125 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
This page was created using
Rationalize Denominator Calculator