Question
Answer
$$\dfrac{7+\sqrt{7}}{8-\sqrt{7}}=\dfrac{21+5\sqrt{7}}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7+\sqrt{7}}{8-\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7+\sqrt{7}}{8-\sqrt{7}}\frac{8+\sqrt{7}}{8+\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{56+7\sqrt{7}+8\sqrt{7}+7}{64+8\sqrt{7}-8\sqrt{7}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{63+15\sqrt{7}}{57} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{21+5\sqrt{7}}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 8 + \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 7 + \sqrt{7}\right) } \cdot \left( 8 + \sqrt{7}\right) = \color{blue}{7} \cdot8+\color{blue}{7} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot8+\color{blue}{ \sqrt{7}} \cdot \sqrt{7} = \\ = 56 + 7 \sqrt{7} + 8 \sqrt{7} + 7 $$ Simplify denominator. $$ \color{blue}{ \left( 8- \sqrt{7}\right) } \cdot \left( 8 + \sqrt{7}\right) = \color{blue}{8} \cdot8+\color{blue}{8} \cdot \sqrt{7}\color{blue}{- \sqrt{7}} \cdot8\color{blue}{- \sqrt{7}} \cdot \sqrt{7} = \\ = 64 + 8 \sqrt{7}- 8 \sqrt{7}-7 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
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