Answer
$$\frac{\sqrt{5}+4}{3-2\sqrt{5}}=-\sqrt{5}-2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}+4}{3-2\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}+4}{3-2\sqrt{5}}\frac{3+2\sqrt{5}}{3+2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3\sqrt{5}+10+12+8\sqrt{5}}{9+6\sqrt{5}-6\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{11\sqrt{5}+22}{-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{5}+2}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{\sqrt{5}+2}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }-(\sqrt{5}+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-\sqrt{5}-2\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 + 2 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{5} + 4\right) } \cdot \left( 3 + 2 \sqrt{5}\right) = \color{blue}{ \sqrt{5}} \cdot3+\color{blue}{ \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{4} \cdot3+\color{blue}{4} \cdot 2 \sqrt{5} = \\ = 3 \sqrt{5} + 10 + 12 + 8 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 3- 2 \sqrt{5}\right) } \cdot \left( 3 + 2 \sqrt{5}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot 2 \sqrt{5}\color{blue}{- 2 \sqrt{5}} \cdot3\color{blue}{- 2 \sqrt{5}} \cdot 2 \sqrt{5} = \\ = 9 + 6 \sqrt{5}- 6 \sqrt{5}-20 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 11. |
| ⑤ | Place a negative sign in front of a fraction. |
| ⑥ | Remove the parenthesis by changing the sign of each term within them. |
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