Answer
$$\frac{4}{3-4\sqrt{5}}=-\frac{12+16\sqrt{5}}{71}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{3-4\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{3-4\sqrt{5}}\frac{3+4\sqrt{5}}{3+4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{12+16\sqrt{5}}{9+12\sqrt{5}-12\sqrt{5}-80} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12+16\sqrt{5}}{-71} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{12+16\sqrt{5}}{71}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 + 4 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( 3 + 4 \sqrt{5}\right) = \color{blue}{4} \cdot3+\color{blue}{4} \cdot 4 \sqrt{5} = \\ = 12 + 16 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 3- 4 \sqrt{5}\right) } \cdot \left( 3 + 4 \sqrt{5}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot 4 \sqrt{5}\color{blue}{- 4 \sqrt{5}} \cdot3\color{blue}{- 4 \sqrt{5}} \cdot 4 \sqrt{5} = \\ = 9 + 12 \sqrt{5}- 12 \sqrt{5}-80 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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