Answer
$$\frac{7}{-2+\sqrt{13}}=\frac{14+7\sqrt{13}}{9}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7}{-2+\sqrt{13}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7}{-2+\sqrt{13}}\frac{-2-\sqrt{13}}{-2-\sqrt{13}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-14-7\sqrt{13}}{4+2\sqrt{13}-2\sqrt{13}-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-14-7\sqrt{13}}{-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{14+7\sqrt{13}}{9}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ -2- \sqrt{13}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 7 } \cdot \left( -2- \sqrt{13}\right) = \color{blue}{7} \cdot-2+\color{blue}{7} \cdot- \sqrt{13} = \\ = -14- 7 \sqrt{13} $$ Simplify denominator. $$ \color{blue}{ \left( -2 + \sqrt{13}\right) } \cdot \left( -2- \sqrt{13}\right) = \color{blue}{-2} \cdot-2\color{blue}{-2} \cdot- \sqrt{13}+\color{blue}{ \sqrt{13}} \cdot-2+\color{blue}{ \sqrt{13}} \cdot- \sqrt{13} = \\ = 4 + 2 \sqrt{13}- 2 \sqrt{13}-13 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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