Question
Answer
$$\dfrac{13}{4-\sqrt{13}}=\dfrac{52+13\sqrt{13}}{3}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{13}{4-\sqrt{13}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{13}{4-\sqrt{13}}\frac{4+\sqrt{13}}{4+\sqrt{13}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{52+13\sqrt{13}}{16+4\sqrt{13}-4\sqrt{13}-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{52+13\sqrt{13}}{3}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 + \sqrt{13}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 13 } \cdot \left( 4 + \sqrt{13}\right) = \color{blue}{13} \cdot4+\color{blue}{13} \cdot \sqrt{13} = \\ = 52 + 13 \sqrt{13} $$ Simplify denominator. $$ \color{blue}{ \left( 4- \sqrt{13}\right) } \cdot \left( 4 + \sqrt{13}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot \sqrt{13}\color{blue}{- \sqrt{13}} \cdot4\color{blue}{- \sqrt{13}} \cdot \sqrt{13} = \\ = 16 + 4 \sqrt{13}- 4 \sqrt{13}-13 $$ |
| ③ | Simplify numerator and denominator |
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