Question
Answer
$$\dfrac{(\sqrt{5}-2)^2}{(\sqrt{5}+2)^2}=161-72\sqrt{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(\sqrt{5}-2)^2}{(\sqrt{5}+2)^2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5-2\sqrt{5}-2\sqrt{5}+4}{5+2\sqrt{5}+2\sqrt{5}+4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9-4\sqrt{5}}{9+4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{9-4\sqrt{5}}{9+4\sqrt{5}}\frac{9-4\sqrt{5}}{9-4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{81-36\sqrt{5}-36\sqrt{5}+80}{81-36\sqrt{5}+36\sqrt{5}-80} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{161-72\sqrt{5}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}161-72\sqrt{5}\end{aligned} $$ | |
| ① | $$ (\sqrt{5}-2)^2 = \left( \sqrt{5}-2 \right) \cdot \left( \sqrt{5}-2 \right) = 5- 2 \sqrt{5}- 2 \sqrt{5} + 4 $$ |
| ② | $$ (\sqrt{5}+2)^2 = \left( \sqrt{5} + 2 \right) \cdot \left( \sqrt{5} + 2 \right) = 5 + 2 \sqrt{5} + 2 \sqrt{5} + 4 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 9- 4 \sqrt{5}} $$. |
| ⑤ | Multiply in a numerator. $$ \color{blue}{ \left( 9- 4 \sqrt{5}\right) } \cdot \left( 9- 4 \sqrt{5}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot- 4 \sqrt{5}\color{blue}{- 4 \sqrt{5}} \cdot9\color{blue}{- 4 \sqrt{5}} \cdot- 4 \sqrt{5} = \\ = 81- 36 \sqrt{5}- 36 \sqrt{5} + 80 $$ Simplify denominator. $$ \color{blue}{ \left( 9 + 4 \sqrt{5}\right) } \cdot \left( 9- 4 \sqrt{5}\right) = \color{blue}{9} \cdot9+\color{blue}{9} \cdot- 4 \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot9+\color{blue}{ 4 \sqrt{5}} \cdot- 4 \sqrt{5} = \\ = 81- 36 \sqrt{5} + 36 \sqrt{5}-80 $$ |
| ⑥ | Simplify numerator and denominator |
| ⑦ | Remove 1 from denominator. |
This page was created using
Rationalize Denominator Calculator