Tap the blue circles to see an explanation.
$$ \begin{aligned}\frac{11+\sqrt{3}}{11-\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{11+\sqrt{3}}{11-\sqrt{3}}\frac{11+\sqrt{3}}{11+\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{121+11\sqrt{3}+11\sqrt{3}+3}{121+11\sqrt{3}-11\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{124+22\sqrt{3}}{118} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{62+11\sqrt{3}}{59}\end{aligned} $$ | |
① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 11 + \sqrt{3}} $$. |
② | Multiply in a numerator. $$ \color{blue}{ \left( 11 + \sqrt{3}\right) } \cdot \left( 11 + \sqrt{3}\right) = \color{blue}{11} \cdot11+\color{blue}{11} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot11+\color{blue}{ \sqrt{3}} \cdot \sqrt{3} = \\ = 121 + 11 \sqrt{3} + 11 \sqrt{3} + 3 $$ Simplify denominator. $$ \color{blue}{ \left( 11- \sqrt{3}\right) } \cdot \left( 11 + \sqrt{3}\right) = \color{blue}{11} \cdot11+\color{blue}{11} \cdot \sqrt{3}\color{blue}{- \sqrt{3}} \cdot11\color{blue}{- \sqrt{3}} \cdot \sqrt{3} = \\ = 121 + 11 \sqrt{3}- 11 \sqrt{3}-3 $$ |
③ | Simplify numerator and denominator |
④ | Divide both numerator and denominator by 2. |