Question
Answer
$$\dfrac{3}{21-\sqrt{19}}=\dfrac{63+3\sqrt{19}}{422}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3}{21-\sqrt{19}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3}{21-\sqrt{19}}\frac{21+\sqrt{19}}{21+\sqrt{19}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{63+3\sqrt{19}}{441+21\sqrt{19}-21\sqrt{19}-19} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{63+3\sqrt{19}}{422}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 21 + \sqrt{19}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 } \cdot \left( 21 + \sqrt{19}\right) = \color{blue}{3} \cdot21+\color{blue}{3} \cdot \sqrt{19} = \\ = 63 + 3 \sqrt{19} $$ Simplify denominator. $$ \color{blue}{ \left( 21- \sqrt{19}\right) } \cdot \left( 21 + \sqrt{19}\right) = \color{blue}{21} \cdot21+\color{blue}{21} \cdot \sqrt{19}\color{blue}{- \sqrt{19}} \cdot21\color{blue}{- \sqrt{19}} \cdot \sqrt{19} = \\ = 441 + 21 \sqrt{19}- 21 \sqrt{19}-19 $$ |
| ③ | Simplify numerator and denominator |
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