Question
Answer
$$\dfrac{7\sqrt{3}}{2\sqrt{3}+6+7\sqrt{3}}=\dfrac{189-42\sqrt{3}}{207}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7\sqrt{3}}{2\sqrt{3}+6+7\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7\sqrt{3}}{9\sqrt{3}+6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{7\sqrt{3}}{9\sqrt{3}+6}\frac{9\sqrt{3}-6}{9\sqrt{3}-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{189-42\sqrt{3}}{243-54\sqrt{3}+54\sqrt{3}-36} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{189-42\sqrt{3}}{207}\end{aligned} $$ | |
| ① | Simplify numerator and denominator |
| ② | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 9 \sqrt{3}-6} $$. |
| ③ | Multiply in a numerator. $$ \color{blue}{ 7 \sqrt{3} } \cdot \left( 9 \sqrt{3}-6\right) = \color{blue}{ 7 \sqrt{3}} \cdot 9 \sqrt{3}+\color{blue}{ 7 \sqrt{3}} \cdot-6 = \\ = 189- 42 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 9 \sqrt{3} + 6\right) } \cdot \left( 9 \sqrt{3}-6\right) = \color{blue}{ 9 \sqrt{3}} \cdot 9 \sqrt{3}+\color{blue}{ 9 \sqrt{3}} \cdot-6+\color{blue}{6} \cdot 9 \sqrt{3}+\color{blue}{6} \cdot-6 = \\ = 243- 54 \sqrt{3} + 54 \sqrt{3}-36 $$ |
| ④ | Simplify numerator and denominator |
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