Answer
$$\frac{-4}{3+\sqrt{13}}=3-\sqrt{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{-4}{3+\sqrt{13}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-4}{3+\sqrt{13}}\frac{3-\sqrt{13}}{3-\sqrt{13}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-12+4\sqrt{13}}{9-3\sqrt{13}+3\sqrt{13}-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-12+4\sqrt{13}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-3+\sqrt{13}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{3-\sqrt{13}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}3-\sqrt{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3- \sqrt{13}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ -4 } \cdot \left( 3- \sqrt{13}\right) = \color{blue}{-4} \cdot3\color{blue}{-4} \cdot- \sqrt{13} = \\ = -12 + 4 \sqrt{13} $$ Simplify denominator. $$ \color{blue}{ \left( 3 + \sqrt{13}\right) } \cdot \left( 3- \sqrt{13}\right) = \color{blue}{3} \cdot3+\color{blue}{3} \cdot- \sqrt{13}+\color{blue}{ \sqrt{13}} \cdot3+\color{blue}{ \sqrt{13}} \cdot- \sqrt{13} = \\ = 9- 3 \sqrt{13} + 3 \sqrt{13}-13 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
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