Answer
$$\frac{2}{\sqrt{3}-4\sqrt{5}}=-\frac{2\sqrt{3}+8\sqrt{5}}{77}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{\sqrt{3}-4\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{\sqrt{3}-4\sqrt{5}}\frac{\sqrt{3}+4\sqrt{5}}{\sqrt{3}+4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{3}+8\sqrt{5}}{3+4\sqrt{15}-4\sqrt{15}-80} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{3}+8\sqrt{5}}{-77} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{2\sqrt{3}+8\sqrt{5}}{77}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{3} + 4 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( \sqrt{3} + 4 \sqrt{5}\right) = \color{blue}{2} \cdot \sqrt{3}+\color{blue}{2} \cdot 4 \sqrt{5} = \\ = 2 \sqrt{3} + 8 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{3}- 4 \sqrt{5}\right) } \cdot \left( \sqrt{3} + 4 \sqrt{5}\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot 4 \sqrt{5}\color{blue}{- 4 \sqrt{5}} \cdot \sqrt{3}\color{blue}{- 4 \sqrt{5}} \cdot 4 \sqrt{5} = \\ = 3 + 4 \sqrt{15}- 4 \sqrt{15}-80 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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