Answer
$$\frac{5\sqrt{6}}{\sqrt{12}-5}=-\frac{30\sqrt{2}+25\sqrt{6}}{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{6}}{\sqrt{12}-5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{6}}{\sqrt{12}-5}\frac{\sqrt{12}+5}{\sqrt{12}+5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30\sqrt{2}+25\sqrt{6}}{12+10\sqrt{3}-10\sqrt{3}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{30\sqrt{2}+25\sqrt{6}}{-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{30\sqrt{2}+25\sqrt{6}}{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{12} + 5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 \sqrt{6} } \cdot \left( \sqrt{12} + 5\right) = \color{blue}{ 5 \sqrt{6}} \cdot \sqrt{12}+\color{blue}{ 5 \sqrt{6}} \cdot5 = \\ = 30 \sqrt{2} + 25 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{12}-5\right) } \cdot \left( \sqrt{12} + 5\right) = \color{blue}{ \sqrt{12}} \cdot \sqrt{12}+\color{blue}{ \sqrt{12}} \cdot5\color{blue}{-5} \cdot \sqrt{12}\color{blue}{-5} \cdot5 = \\ = 12 + 10 \sqrt{3}- 10 \sqrt{3}-25 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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