Question
Answer
$$\dfrac{1}{\sqrt{5}-2\sqrt{2}}=-\dfrac{\sqrt{5}+2\sqrt{2}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{\sqrt{5}-2\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{\sqrt{5}-2\sqrt{2}}\frac{\sqrt{5}+2\sqrt{2}}{\sqrt{5}+2\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{5}+2\sqrt{2}}{5+2\sqrt{10}-2\sqrt{10}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{5}+2\sqrt{2}}{-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{\sqrt{5}+2\sqrt{2}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5} + 2 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( \sqrt{5} + 2 \sqrt{2}\right) = \color{blue}{1} \cdot \sqrt{5}+\color{blue}{1} \cdot 2 \sqrt{2} = \\ = \sqrt{5} + 2 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5}- 2 \sqrt{2}\right) } \cdot \left( \sqrt{5} + 2 \sqrt{2}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot 2 \sqrt{2}\color{blue}{- 2 \sqrt{2}} \cdot \sqrt{5}\color{blue}{- 2 \sqrt{2}} \cdot 2 \sqrt{2} = \\ = 5 + 2 \sqrt{10}- 2 \sqrt{10}-8 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
This page was created using
Rationalize Denominator Calculator