Question
Answer
$$\dfrac{6}{4\sqrt{6}-6\sqrt{3}}=-2\sqrt{6}-3\sqrt{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6}{4\sqrt{6}-6\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6}{4\sqrt{6}-6\sqrt{3}}\frac{4\sqrt{6}+6\sqrt{3}}{4\sqrt{6}+6\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{24\sqrt{6}+36\sqrt{3}}{96+72\sqrt{2}-72\sqrt{2}-108} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{24\sqrt{6}+36\sqrt{3}}{-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{2\sqrt{6}+3\sqrt{3}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{2\sqrt{6}+3\sqrt{3}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }-(2\sqrt{6}+3\sqrt{3}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-2\sqrt{6}-3\sqrt{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{6} + 6 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 } \cdot \left( 4 \sqrt{6} + 6 \sqrt{3}\right) = \color{blue}{6} \cdot 4 \sqrt{6}+\color{blue}{6} \cdot 6 \sqrt{3} = \\ = 24 \sqrt{6} + 36 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{6}- 6 \sqrt{3}\right) } \cdot \left( 4 \sqrt{6} + 6 \sqrt{3}\right) = \color{blue}{ 4 \sqrt{6}} \cdot 4 \sqrt{6}+\color{blue}{ 4 \sqrt{6}} \cdot 6 \sqrt{3}\color{blue}{- 6 \sqrt{3}} \cdot 4 \sqrt{6}\color{blue}{- 6 \sqrt{3}} \cdot 6 \sqrt{3} = \\ = 96 + 72 \sqrt{2}- 72 \sqrt{2}-108 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 12. |
| ⑤ | Place a negative sign in front of a fraction. |
| ⑥ | Remove the parenthesis by changing the sign of each term within them. |
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