Question
Answer
$$\dfrac{\sqrt{6}-2\sqrt{3}}{\sqrt{6}+2\sqrt{3}}=-3+2\sqrt{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{6}-2\sqrt{3}}{\sqrt{6}+2\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{6}-2\sqrt{3}}{\sqrt{6}+2\sqrt{3}}\frac{\sqrt{6}-2\sqrt{3}}{\sqrt{6}-2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6-6\sqrt{2}-6\sqrt{2}+12}{6-6\sqrt{2}+6\sqrt{2}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{18-12\sqrt{2}}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3-2\sqrt{2}}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-3+2\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-3+2\sqrt{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{6}- 2 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{6}- 2 \sqrt{3}\right) } \cdot \left( \sqrt{6}- 2 \sqrt{3}\right) = \color{blue}{ \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot- 2 \sqrt{3}\color{blue}{- 2 \sqrt{3}} \cdot \sqrt{6}\color{blue}{- 2 \sqrt{3}} \cdot- 2 \sqrt{3} = \\ = 6- 6 \sqrt{2}- 6 \sqrt{2} + 12 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{6} + 2 \sqrt{3}\right) } \cdot \left( \sqrt{6}- 2 \sqrt{3}\right) = \color{blue}{ \sqrt{6}} \cdot \sqrt{6}+\color{blue}{ \sqrt{6}} \cdot- 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot \sqrt{6}+\color{blue}{ 2 \sqrt{3}} \cdot- 2 \sqrt{3} = \\ = 6- 6 \sqrt{2} + 6 \sqrt{2}-12 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 6. |
| ⑤ | Multiply both numerator and denominator by -1. |
| ⑥ | Remove 1 from denominator. |
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