Answer
$$\frac{1}{12-\sqrt{11}}=\frac{12+\sqrt{11}}{133}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{12-\sqrt{11}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{12-\sqrt{11}}\frac{12+\sqrt{11}}{12+\sqrt{11}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{12+\sqrt{11}}{144+12\sqrt{11}-12\sqrt{11}-11} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12+\sqrt{11}}{133}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 12 + \sqrt{11}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 1 } \cdot \left( 12 + \sqrt{11}\right) = \color{blue}{1} \cdot12+\color{blue}{1} \cdot \sqrt{11} = \\ = 12 + \sqrt{11} $$ Simplify denominator. $$ \color{blue}{ \left( 12- \sqrt{11}\right) } \cdot \left( 12 + \sqrt{11}\right) = \color{blue}{12} \cdot12+\color{blue}{12} \cdot \sqrt{11}\color{blue}{- \sqrt{11}} \cdot12\color{blue}{- \sqrt{11}} \cdot \sqrt{11} = \\ = 144 + 12 \sqrt{11}- 12 \sqrt{11}-11 $$ |
| ③ | Simplify numerator and denominator |
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