Answer
$$\frac{\sqrt{7}}{7-\sqrt{7}}=\frac{\sqrt{7}+1}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{7}}{7-\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{7}}{7-\sqrt{7}}\frac{7+\sqrt{7}}{7+\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{7\sqrt{7}+7}{49+7\sqrt{7}-7\sqrt{7}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{7\sqrt{7}+7}{42} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{7}+1}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 7 + \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{7} } \cdot \left( 7 + \sqrt{7}\right) = \color{blue}{ \sqrt{7}} \cdot7+\color{blue}{ \sqrt{7}} \cdot \sqrt{7} = \\ = 7 \sqrt{7} + 7 $$ Simplify denominator. $$ \color{blue}{ \left( 7- \sqrt{7}\right) } \cdot \left( 7 + \sqrt{7}\right) = \color{blue}{7} \cdot7+\color{blue}{7} \cdot \sqrt{7}\color{blue}{- \sqrt{7}} \cdot7\color{blue}{- \sqrt{7}} \cdot \sqrt{7} = \\ = 49 + 7 \sqrt{7}- 7 \sqrt{7}-7 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 7. |
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