Question
Answer
$$\dfrac{3+5\sqrt{19}}{7-\sqrt{19}}=\dfrac{58+19\sqrt{19}}{15}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3+5\sqrt{19}}{7-\sqrt{19}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3+5\sqrt{19}}{7-\sqrt{19}}\frac{7+\sqrt{19}}{7+\sqrt{19}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{21+3\sqrt{19}+35\sqrt{19}+95}{49+7\sqrt{19}-7\sqrt{19}-19} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{116+38\sqrt{19}}{30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{58+19\sqrt{19}}{15}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7 + \sqrt{19}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 + 5 \sqrt{19}\right) } \cdot \left( 7 + \sqrt{19}\right) = \color{blue}{3} \cdot7+\color{blue}{3} \cdot \sqrt{19}+\color{blue}{ 5 \sqrt{19}} \cdot7+\color{blue}{ 5 \sqrt{19}} \cdot \sqrt{19} = \\ = 21 + 3 \sqrt{19} + 35 \sqrt{19} + 95 $$ Simplify denominator. $$ \color{blue}{ \left( 7- \sqrt{19}\right) } \cdot \left( 7 + \sqrt{19}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot \sqrt{19}\color{blue}{- \sqrt{19}} \cdot7\color{blue}{- \sqrt{19}} \cdot \sqrt{19} = \\ = 49 + 7 \sqrt{19}- 7 \sqrt{19}-19 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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