Question
Answer
$$\dfrac{7+6\sqrt{3}}{7-7\sqrt{3}}=-\dfrac{25+13\sqrt{3}}{14}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7+6\sqrt{3}}{7-7\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7+6\sqrt{3}}{7-7\sqrt{3}}\frac{7+7\sqrt{3}}{7+7\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{49+49\sqrt{3}+42\sqrt{3}+126}{49+49\sqrt{3}-49\sqrt{3}-147} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{175+91\sqrt{3}}{-98} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{25+13\sqrt{3}}{-14} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{25+13\sqrt{3}}{14}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7 + 7 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 7 + 6 \sqrt{3}\right) } \cdot \left( 7 + 7 \sqrt{3}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot 7 \sqrt{3}+\color{blue}{ 6 \sqrt{3}} \cdot7+\color{blue}{ 6 \sqrt{3}} \cdot 7 \sqrt{3} = \\ = 49 + 49 \sqrt{3} + 42 \sqrt{3} + 126 $$ Simplify denominator. $$ \color{blue}{ \left( 7- 7 \sqrt{3}\right) } \cdot \left( 7 + 7 \sqrt{3}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot 7 \sqrt{3}\color{blue}{- 7 \sqrt{3}} \cdot7\color{blue}{- 7 \sqrt{3}} \cdot 7 \sqrt{3} = \\ = 49 + 49 \sqrt{3}- 49 \sqrt{3}-147 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 7. |
| ⑤ | Place a negative sign in front of a fraction. |
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