Tap the blue circles to see an explanation.
$$ \begin{aligned}\frac{\sqrt{11}-\sqrt{3}}{\sqrt{11}+\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{11}-\sqrt{3}}{\sqrt{11}+\sqrt{3}}\frac{\sqrt{11}-\sqrt{3}}{\sqrt{11}-\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{11-\sqrt{33}-\sqrt{33}+3}{11-\sqrt{33}+\sqrt{33}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{14-2\sqrt{33}}{8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{7-\sqrt{33}}{4}\end{aligned} $$ | |
① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{11}- \sqrt{3}} $$. |
② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{11}- \sqrt{3}\right) } \cdot \left( \sqrt{11}- \sqrt{3}\right) = \color{blue}{ \sqrt{11}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot- \sqrt{3}\color{blue}{- \sqrt{3}} \cdot \sqrt{11}\color{blue}{- \sqrt{3}} \cdot- \sqrt{3} = \\ = 11- \sqrt{33}- \sqrt{33} + 3 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{11} + \sqrt{3}\right) } \cdot \left( \sqrt{11}- \sqrt{3}\right) = \color{blue}{ \sqrt{11}} \cdot \sqrt{11}+\color{blue}{ \sqrt{11}} \cdot- \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot \sqrt{11}+\color{blue}{ \sqrt{3}} \cdot- \sqrt{3} = \\ = 11- \sqrt{33} + \sqrt{33}-3 $$ |
③ | Simplify numerator and denominator |
④ | Divide both numerator and denominator by 2. |