Question
Answer
$$\dfrac{2+\sqrt{12}}{\sqrt{3}+\sqrt{5}}=-\sqrt{3}+\sqrt{5}-3+\sqrt{15}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2+\sqrt{12}}{\sqrt{3}+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2+\sqrt{12}}{\sqrt{3}+\sqrt{5}}\frac{\sqrt{3}-\sqrt{5}}{\sqrt{3}-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2\sqrt{3}-2\sqrt{5}+6-2\sqrt{15}}{3-\sqrt{15}+\sqrt{15}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2\sqrt{3}-2\sqrt{5}+6-2\sqrt{15}}{-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{3}-\sqrt{5}+3-\sqrt{15}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-\sqrt{3}+\sqrt{5}-3+\sqrt{15}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-\sqrt{3}+\sqrt{5}-3+\sqrt{15}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{3}- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 + \sqrt{12}\right) } \cdot \left( \sqrt{3}- \sqrt{5}\right) = \color{blue}{2} \cdot \sqrt{3}+\color{blue}{2} \cdot- \sqrt{5}+\color{blue}{ \sqrt{12}} \cdot \sqrt{3}+\color{blue}{ \sqrt{12}} \cdot- \sqrt{5} = \\ = 2 \sqrt{3}- 2 \sqrt{5} + 6- 2 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{3} + \sqrt{5}\right) } \cdot \left( \sqrt{3}- \sqrt{5}\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot \sqrt{3}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 3- \sqrt{15} + \sqrt{15}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
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