Answer
$$\frac{7}{11-\sqrt{7}}=\frac{77+7\sqrt{7}}{114}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7}{11-\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7}{11-\sqrt{7}}\frac{11+\sqrt{7}}{11+\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{77+7\sqrt{7}}{121+11\sqrt{7}-11\sqrt{7}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{77+7\sqrt{7}}{114}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 11 + \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 7 } \cdot \left( 11 + \sqrt{7}\right) = \color{blue}{7} \cdot11+\color{blue}{7} \cdot \sqrt{7} = \\ = 77 + 7 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( 11- \sqrt{7}\right) } \cdot \left( 11 + \sqrt{7}\right) = \color{blue}{11} \cdot11+\color{blue}{11} \cdot \sqrt{7}\color{blue}{- \sqrt{7}} \cdot11\color{blue}{- \sqrt{7}} \cdot \sqrt{7} = \\ = 121 + 11 \sqrt{7}- 11 \sqrt{7}-7 $$ |
| ③ | Simplify numerator and denominator |
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