Question
Answer
$$\dfrac{\sqrt{11}-\sqrt{3}}{11+\sqrt{3}}=\dfrac{11\sqrt{11}-\sqrt{33}-11\sqrt{3}+3}{118}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{11}-\sqrt{3}}{11+\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{11}-\sqrt{3}}{11+\sqrt{3}}\frac{11-\sqrt{3}}{11-\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{11\sqrt{11}-\sqrt{33}-11\sqrt{3}+3}{121-11\sqrt{3}+11\sqrt{3}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{11\sqrt{11}-\sqrt{33}-11\sqrt{3}+3}{118}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 11- \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{11}- \sqrt{3}\right) } \cdot \left( 11- \sqrt{3}\right) = \color{blue}{ \sqrt{11}} \cdot11+\color{blue}{ \sqrt{11}} \cdot- \sqrt{3}\color{blue}{- \sqrt{3}} \cdot11\color{blue}{- \sqrt{3}} \cdot- \sqrt{3} = \\ = 11 \sqrt{11}- \sqrt{33}- 11 \sqrt{3} + 3 $$ 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 |
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