Question
Answer
$$\dfrac{\sqrt{3}}{6-6\sqrt{2}}=-\dfrac{\sqrt{3}+\sqrt{6}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{3}}{6-6\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{3}}{6-6\sqrt{2}}\frac{6+6\sqrt{2}}{6+6\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{3}+6\sqrt{6}}{36+36\sqrt{2}-36\sqrt{2}-72} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6\sqrt{3}+6\sqrt{6}}{-36} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{3}+\sqrt{6}}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{\sqrt{3}+\sqrt{6}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6 + 6 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{3} } \cdot \left( 6 + 6 \sqrt{2}\right) = \color{blue}{ \sqrt{3}} \cdot6+\color{blue}{ \sqrt{3}} \cdot 6 \sqrt{2} = \\ = 6 \sqrt{3} + 6 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 6- 6 \sqrt{2}\right) } \cdot \left( 6 + 6 \sqrt{2}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot 6 \sqrt{2}\color{blue}{- 6 \sqrt{2}} \cdot6\color{blue}{- 6 \sqrt{2}} \cdot 6 \sqrt{2} = \\ = 36 + 36 \sqrt{2}- 36 \sqrt{2}-72 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 6. |
| ⑤ | Place a negative sign in front of a fraction. |
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