Question
Answer
$$\dfrac{13}{19+8\sqrt{3}}=\dfrac{19-8\sqrt{3}}{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{13}{19+8\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{13}{19+8\sqrt{3}}\frac{19-8\sqrt{3}}{19-8\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{247-104\sqrt{3}}{361-152\sqrt{3}+152\sqrt{3}-192} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{247-104\sqrt{3}}{169} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{19-8\sqrt{3}}{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 19- 8 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 13 } \cdot \left( 19- 8 \sqrt{3}\right) = \color{blue}{13} \cdot19+\color{blue}{13} \cdot- 8 \sqrt{3} = \\ = 247- 104 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 19 + 8 \sqrt{3}\right) } \cdot \left( 19- 8 \sqrt{3}\right) = \color{blue}{19} \cdot19+\color{blue}{19} \cdot- 8 \sqrt{3}+\color{blue}{ 8 \sqrt{3}} \cdot19+\color{blue}{ 8 \sqrt{3}} \cdot- 8 \sqrt{3} = \\ = 361- 152 \sqrt{3} + 152 \sqrt{3}-192 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 13. |
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