Answer
$$\frac{\sqrt{13}+9}{\sqrt{12}-18}=-\frac{\sqrt{39}+9\sqrt{13}+9\sqrt{3}+81}{156}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{13}+9}{\sqrt{12}-18}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{13}+9}{\sqrt{12}-18}\frac{\sqrt{12}+18}{\sqrt{12}+18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{39}+18\sqrt{13}+18\sqrt{3}+162}{12+36\sqrt{3}-36\sqrt{3}-324} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{39}+18\sqrt{13}+18\sqrt{3}+162}{-312} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{39}+9\sqrt{13}+9\sqrt{3}+81}{-156} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{\sqrt{39}+9\sqrt{13}+9\sqrt{3}+81}{156}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{12} + 18} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{13} + 9\right) } \cdot \left( \sqrt{12} + 18\right) = \color{blue}{ \sqrt{13}} \cdot \sqrt{12}+\color{blue}{ \sqrt{13}} \cdot18+\color{blue}{9} \cdot \sqrt{12}+\color{blue}{9} \cdot18 = \\ = 2 \sqrt{39} + 18 \sqrt{13} + 18 \sqrt{3} + 162 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{12}-18\right) } \cdot \left( \sqrt{12} + 18\right) = \color{blue}{ \sqrt{12}} \cdot \sqrt{12}+\color{blue}{ \sqrt{12}} \cdot18\color{blue}{-18} \cdot \sqrt{12}\color{blue}{-18} \cdot18 = \\ = 12 + 36 \sqrt{3}- 36 \sqrt{3}-324 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Place a negative sign in front of a fraction. |
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