Question
Answer
$$\dfrac{6}{13-2\sqrt{13}}=\dfrac{78+12\sqrt{13}}{117}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{6}{13-2\sqrt{13}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6}{13-2\sqrt{13}}\frac{13+2\sqrt{13}}{13+2\sqrt{13}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{78+12\sqrt{13}}{169+26\sqrt{13}-26\sqrt{13}-52} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{78+12\sqrt{13}}{117}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 13 + 2 \sqrt{13}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 6 } \cdot \left( 13 + 2 \sqrt{13}\right) = \color{blue}{6} \cdot13+\color{blue}{6} \cdot 2 \sqrt{13} = \\ = 78 + 12 \sqrt{13} $$ Simplify denominator. $$ \color{blue}{ \left( 13- 2 \sqrt{13}\right) } \cdot \left( 13 + 2 \sqrt{13}\right) = \color{blue}{13} \cdot13+\color{blue}{13} \cdot 2 \sqrt{13}\color{blue}{- 2 \sqrt{13}} \cdot13\color{blue}{- 2 \sqrt{13}} \cdot 2 \sqrt{13} = \\ = 169 + 26 \sqrt{13}- 26 \sqrt{13}-52 $$ |
| ③ | Simplify numerator and denominator |
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