Answer
$$\frac{(11+\sqrt{3})\cdot(11-\sqrt{3})}{(11-\sqrt{3})\cdot(11+\sqrt{3})}=1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(11+\sqrt{3})\cdot(11-\sqrt{3})}{(11-\sqrt{3})\cdot(11+\sqrt{3})}& \xlongequal{ }(121-11\sqrt{3}+11\sqrt{3}-3)\cdot\frac{1}{121+11\sqrt{3}-11\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{121-11\sqrt{3}+11\sqrt{3}-3}{121+11\sqrt{3}-11\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{118}{118} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}} \frac{ 118 : \color{orangered}{ 118 } }{ 118 : \color{orangered}{ 118 }} \xlongequal{ } \\[1 em] & \xlongequal{ }\frac{1}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}1\end{aligned} $$ | |
| ① | $$ \color{blue}{ \left( 121- 11 \sqrt{3} + 11 \sqrt{3}-3\right) } \cdot 1 = \color{blue}{121} \cdot1\color{blue}{- 11 \sqrt{3}} \cdot1+\color{blue}{ 11 \sqrt{3}} \cdot1\color{blue}{-3} \cdot1 = \\ = 121- 11 \sqrt{3} + 11 \sqrt{3}-3 $$$$ \color{blue}{ 1 } \cdot \left( 121 + 11 \sqrt{3}- 11 \sqrt{3}-3\right) = \color{blue}{1} \cdot121+\color{blue}{1} \cdot 11 \sqrt{3}+\color{blue}{1} \cdot- 11 \sqrt{3}+\color{blue}{1} \cdot-3 = \\ = 121 + 11 \sqrt{3}- 11 \sqrt{3}-3 $$ |
| ② | Simplify numerator and denominator |
| ③ | Divide both the top and bottom numbers by $ \color{orangered}{ 118 } $. |
| ④ | Remove 1 from denominator. |
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