Answer
$$\frac{5}{6-\sqrt{3}\cdot7}=-\frac{30+35\sqrt{3}}{111}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5}{6-\sqrt{3}\cdot7}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5}{6-\sqrt{3}\cdot7}\frac{6+7\sqrt{3}}{6+7\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30+35\sqrt{3}}{36+42\sqrt{3}-42\sqrt{3}-147} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{30+35\sqrt{3}}{-111} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{30+35\sqrt{3}}{111}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6 + 7 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 5 } \cdot \left( 6 + 7 \sqrt{3}\right) = \color{blue}{5} \cdot6+\color{blue}{5} \cdot 7 \sqrt{3} = \\ = 30 + 35 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 6- 7 \sqrt{3}\right) } \cdot \left( 6 + 7 \sqrt{3}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot 7 \sqrt{3}\color{blue}{- 7 \sqrt{3}} \cdot6\color{blue}{- 7 \sqrt{3}} \cdot 7 \sqrt{3} = \\ = 36 + 42 \sqrt{3}- 42 \sqrt{3}-147 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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