Question
Answer
$$\dfrac{3}{4+4\sqrt{5}}=\dfrac{-12+12\sqrt{5}}{64}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3}{4+4\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3}{4+4\sqrt{5}}\frac{4-4\sqrt{5}}{4-4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{12-12\sqrt{5}}{16-16\sqrt{5}+16\sqrt{5}-80} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12-12\sqrt{5}}{-64} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-12+12\sqrt{5}}{64}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4- 4 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 } \cdot \left( 4- 4 \sqrt{5}\right) = \color{blue}{3} \cdot4+\color{blue}{3} \cdot- 4 \sqrt{5} = \\ = 12- 12 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 4 + 4 \sqrt{5}\right) } \cdot \left( 4- 4 \sqrt{5}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot- 4 \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot4+\color{blue}{ 4 \sqrt{5}} \cdot- 4 \sqrt{5} = \\ = 16- 16 \sqrt{5} + 16 \sqrt{5}-80 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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