Answer
$$\frac{2}{13+7\sqrt{3}}=\frac{13-7\sqrt{3}}{11}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2}{13+7\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{13+7\sqrt{3}}\frac{13-7\sqrt{3}}{13-7\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{26-14\sqrt{3}}{169-91\sqrt{3}+91\sqrt{3}-147} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{26-14\sqrt{3}}{22} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{13-7\sqrt{3}}{11}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 13- 7 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 2 } \cdot \left( 13- 7 \sqrt{3}\right) = \color{blue}{2} \cdot13+\color{blue}{2} \cdot- 7 \sqrt{3} = \\ = 26- 14 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 13 + 7 \sqrt{3}\right) } \cdot \left( 13- 7 \sqrt{3}\right) = \color{blue}{13} \cdot13+\color{blue}{13} \cdot- 7 \sqrt{3}+\color{blue}{ 7 \sqrt{3}} \cdot13+\color{blue}{ 7 \sqrt{3}} \cdot- 7 \sqrt{3} = \\ = 169- 91 \sqrt{3} + 91 \sqrt{3}-147 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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