Answer
$$\frac{4}{4-2\sqrt{5}}=-\frac{16+8\sqrt{5}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{4-2\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{4-2\sqrt{5}}\frac{4+2\sqrt{5}}{4+2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{16+8\sqrt{5}}{16+8\sqrt{5}-8\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{16+8\sqrt{5}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{16+8\sqrt{5}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 + 2 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( 4 + 2 \sqrt{5}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot 2 \sqrt{5} = \\ = 16 + 8 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 4- 2 \sqrt{5}\right) } \cdot \left( 4 + 2 \sqrt{5}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot 2 \sqrt{5}\color{blue}{- 2 \sqrt{5}} \cdot4\color{blue}{- 2 \sqrt{5}} \cdot 2 \sqrt{5} = \\ = 16 + 8 \sqrt{5}- 8 \sqrt{5}-20 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
This page was created using
Rationalize Denominator Calculator