Question
Answer
$$\dfrac{1}{2-2\sqrt{3}}=-\dfrac{1+\sqrt{3}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{2-2\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{2-2\sqrt{3}}\frac{2+2\sqrt{3}}{2+2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2+2\sqrt{3}}{4+4\sqrt{3}-4\sqrt{3}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2+2\sqrt{3}}{-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1+\sqrt{3}}{-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{1+\sqrt{3}}{4}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 + 2 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 2 + 2 \sqrt{3}\right) = \color{blue}{1} \cdot2+\color{blue}{1} \cdot 2 \sqrt{3} = \\ = 2 + 2 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 2- 2 \sqrt{3}\right) } \cdot \left( 2 + 2 \sqrt{3}\right) = \color{blue}{2} \cdot2+\color{blue}{2} \cdot 2 \sqrt{3}\color{blue}{- 2 \sqrt{3}} \cdot2\color{blue}{- 2 \sqrt{3}} \cdot 2 \sqrt{3} = \\ = 4 + 4 \sqrt{3}- 4 \sqrt{3}-12 $$ |
| 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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