Question
Answer
$$\dfrac{2\sqrt{6}-3\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}=-1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{6}-3\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{6}-3\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}\frac{3\sqrt{5}+2\sqrt{6}}{3\sqrt{5}+2\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{30}+24-45-6\sqrt{30}}{45+6\sqrt{30}-6\sqrt{30}-24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-21}{21} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{21}{21} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}- \, \frac{ 21 : \color{orangered}{ 21 } }{ 21 : \color{orangered}{ 21 }} \xlongequal{ } \\[1 em] & \xlongequal{ }-\frac{1}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-1\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{5} + 2 \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 \sqrt{6}- 3 \sqrt{5}\right) } \cdot \left( 3 \sqrt{5} + 2 \sqrt{6}\right) = \color{blue}{ 2 \sqrt{6}} \cdot 3 \sqrt{5}+\color{blue}{ 2 \sqrt{6}} \cdot 2 \sqrt{6}\color{blue}{- 3 \sqrt{5}} \cdot 3 \sqrt{5}\color{blue}{- 3 \sqrt{5}} \cdot 2 \sqrt{6} = \\ = 6 \sqrt{30} + 24-45- 6 \sqrt{30} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{5}- 2 \sqrt{6}\right) } \cdot \left( 3 \sqrt{5} + 2 \sqrt{6}\right) = \color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot 2 \sqrt{6}\color{blue}{- 2 \sqrt{6}} \cdot 3 \sqrt{5}\color{blue}{- 2 \sqrt{6}} \cdot 2 \sqrt{6} = \\ = 45 + 6 \sqrt{30}- 6 \sqrt{30}-24 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place minus sign in front of the fraction. |
| ⑤ | Divide both the top and bottom numbers by $ \color{orangered}{ 21 } $. |
| ⑥ | Remove 1 from denominator. |
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