Question
Answer
$$\dfrac{13}{5-2\sqrt{3}}=5+2\sqrt{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{13}{5-2\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{13}{5-2\sqrt{3}}\frac{5+2\sqrt{3}}{5+2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{65+26\sqrt{3}}{25+10\sqrt{3}-10\sqrt{3}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{65+26\sqrt{3}}{13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5+2\sqrt{3}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}5+2\sqrt{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 + 2 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 13 } \cdot \left( 5 + 2 \sqrt{3}\right) = \color{blue}{13} \cdot5+\color{blue}{13} \cdot 2 \sqrt{3} = \\ = 65 + 26 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( 5- 2 \sqrt{3}\right) } \cdot \left( 5 + 2 \sqrt{3}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot 2 \sqrt{3}\color{blue}{- 2 \sqrt{3}} \cdot5\color{blue}{- 2 \sqrt{3}} \cdot 2 \sqrt{3} = \\ = 25 + 10 \sqrt{3}- 10 \sqrt{3}-12 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 13. |
| ⑤ | Remove 1 from denominator. |
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