Answer
$$\frac{6}{7+\sqrt{13}}=\frac{7-\sqrt{13}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6}{7+\sqrt{13}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6}{7+\sqrt{13}}\frac{7-\sqrt{13}}{7-\sqrt{13}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{42-6\sqrt{13}}{49-7\sqrt{13}+7\sqrt{13}-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{42-6\sqrt{13}}{36} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{7-\sqrt{13}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7- \sqrt{13}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 } \cdot \left( 7- \sqrt{13}\right) = \color{blue}{6} \cdot7+\color{blue}{6} \cdot- \sqrt{13} = \\ = 42- 6 \sqrt{13} $$ Simplify denominator. $$ \color{blue}{ \left( 7 + \sqrt{13}\right) } \cdot \left( 7- \sqrt{13}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot- \sqrt{13}+\color{blue}{ \sqrt{13}} \cdot7+\color{blue}{ \sqrt{13}} \cdot- \sqrt{13} = \\ = 49- 7 \sqrt{13} + 7 \sqrt{13}-13 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 6. |
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