Answer
$$\frac{\sqrt{13}}{\sqrt{2}-4\sqrt{5}}=-\frac{\sqrt{26}+4\sqrt{65}}{78}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{13}}{\sqrt{2}-4\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{13}}{\sqrt{2}-4\sqrt{5}}\frac{\sqrt{2}+4\sqrt{5}}{\sqrt{2}+4\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{26}+4\sqrt{65}}{2+4\sqrt{10}-4\sqrt{10}-80} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{26}+4\sqrt{65}}{-78} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{\sqrt{26}+4\sqrt{65}}{78}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{2} + 4 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{13} } \cdot \left( \sqrt{2} + 4 \sqrt{5}\right) = \color{blue}{ \sqrt{13}} \cdot \sqrt{2}+\color{blue}{ \sqrt{13}} \cdot 4 \sqrt{5} = \\ = \sqrt{26} + 4 \sqrt{65} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{2}- 4 \sqrt{5}\right) } \cdot \left( \sqrt{2} + 4 \sqrt{5}\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot 4 \sqrt{5}\color{blue}{- 4 \sqrt{5}} \cdot \sqrt{2}\color{blue}{- 4 \sqrt{5}} \cdot 4 \sqrt{5} = \\ = 2 + 4 \sqrt{10}- 4 \sqrt{10}-80 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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