Answer
$$\frac{7}{11+\sqrt{11}}=\frac{77-7\sqrt{11}}{110}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7}{11+\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7}{11+\sqrt{11}}\frac{11-\sqrt{11}}{11-\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{77-7\sqrt{11}}{121-11\sqrt{11}+11\sqrt{11}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{77-7\sqrt{11}}{110}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 11- \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 7 } \cdot \left( 11- \sqrt{11}\right) = \color{blue}{7} \cdot11+\color{blue}{7} \cdot- \sqrt{11} = \\ = 77- 7 \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ \left( 11 + \sqrt{11}\right) } \cdot \left( 11- \sqrt{11}\right) = \color{blue}{11} \cdot11+\color{blue}{11} \cdot- \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot11+\color{blue}{ \sqrt{11}} \cdot- \sqrt{11} = \\ = 121- 11 \sqrt{11} + 11 \sqrt{11}-11 $$ |
| ③ | Simplify numerator and denominator |
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