Question
Answer
$$\dfrac{21+\sqrt{79}}{21-\sqrt{79}}=\dfrac{260+21\sqrt{79}}{181}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{21+\sqrt{79}}{21-\sqrt{79}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{21+\sqrt{79}}{21-\sqrt{79}}\frac{21+\sqrt{79}}{21+\sqrt{79}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{441+21\sqrt{79}+21\sqrt{79}+79}{441+21\sqrt{79}-21\sqrt{79}-79} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{520+42\sqrt{79}}{362} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{260+21\sqrt{79}}{181}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 21 + \sqrt{79}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 21 + \sqrt{79}\right) } \cdot \left( 21 + \sqrt{79}\right) = \color{blue}{21} \cdot21+\color{blue}{21} \cdot \sqrt{79}+\color{blue}{ \sqrt{79}} \cdot21+\color{blue}{ \sqrt{79}} \cdot \sqrt{79} = \\ = 441 + 21 \sqrt{79} + 21 \sqrt{79} + 79 $$ Simplify denominator. $$ \color{blue}{ \left( 21- \sqrt{79}\right) } \cdot \left( 21 + \sqrt{79}\right) = \color{blue}{21} \cdot21+\color{blue}{21} \cdot \sqrt{79}\color{blue}{- \sqrt{79}} \cdot21\color{blue}{- \sqrt{79}} \cdot \sqrt{79} = \\ = 441 + 21 \sqrt{79}- 21 \sqrt{79}-79 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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