Question
Answer
$$\dfrac{\sqrt{2}+1}{\sqrt{3}-2}=-\sqrt{6}-2\sqrt{2}-\sqrt{3}-2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}+1}{\sqrt{3}-2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}+1}{\sqrt{3}-2}\frac{\sqrt{3}+2}{\sqrt{3}+2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\sqrt{6}+2\sqrt{2}+\sqrt{3}+2}{3+2\sqrt{3}-2\sqrt{3}-4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\sqrt{6}+2\sqrt{2}+\sqrt{3}+2}{-1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{\sqrt{6}+2\sqrt{2}+\sqrt{3}+2}{1} \xlongequal{ } \\[1 em] & \xlongequal{ }-(\sqrt{6}+2\sqrt{2}+\sqrt{3}+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\sqrt{6}-2\sqrt{2}-\sqrt{3}-2\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{3} + 2} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{2} + 1\right) } \cdot \left( \sqrt{3} + 2\right) = \color{blue}{ \sqrt{2}} \cdot \sqrt{3}+\color{blue}{ \sqrt{2}} \cdot2+\color{blue}{1} \cdot \sqrt{3}+\color{blue}{1} \cdot2 = \\ = \sqrt{6} + 2 \sqrt{2} + \sqrt{3} + 2 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{3}-2\right) } \cdot \left( \sqrt{3} + 2\right) = \color{blue}{ \sqrt{3}} \cdot \sqrt{3}+\color{blue}{ \sqrt{3}} \cdot2\color{blue}{-2} \cdot \sqrt{3}\color{blue}{-2} \cdot2 = \\ = 3 + 2 \sqrt{3}- 2 \sqrt{3}-4 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
| ⑤ | Remove the parenthesis by changing the sign of each term within them. |
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