Question
Answer
$$\dfrac{6}{1-5\sqrt{3}}=-\dfrac{6+30\sqrt{3}}{74}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6}{1-5\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6}{1-5\sqrt{3}}\frac{1+5\sqrt{3}}{1+5\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6+30\sqrt{3}}{1+5\sqrt{3}-5\sqrt{3}-75} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6+30\sqrt{3}}{-74} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{6+30\sqrt{3}}{74}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 1 + 5 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 } \cdot \left( 1 + 5 \sqrt{3}\right) = \color{blue}{6} \cdot1+\color{blue}{6} \cdot 5 \sqrt{3} = \\ = 6 + 30 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 1- 5 \sqrt{3}\right) } \cdot \left( 1 + 5 \sqrt{3}\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot 5 \sqrt{3}\color{blue}{- 5 \sqrt{3}} \cdot1\color{blue}{- 5 \sqrt{3}} \cdot 5 \sqrt{3} = \\ = 1 + 5 \sqrt{3}- 5 \sqrt{3}-75 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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