Question
Answer
$$\dfrac{2\sqrt{5}-3\sqrt{6}}{2\sqrt{5}+3\sqrt{6}}=\dfrac{-37+6\sqrt{30}}{17}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{2\sqrt{5}-3\sqrt{6}}{2\sqrt{5}+3\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2\sqrt{5}-3\sqrt{6}}{2\sqrt{5}+3\sqrt{6}}\frac{2\sqrt{5}-3\sqrt{6}}{2\sqrt{5}-3\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{20-6\sqrt{30}-6\sqrt{30}+54}{20-6\sqrt{30}+6\sqrt{30}-54} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{74-12\sqrt{30}}{-34} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{37-6\sqrt{30}}{-17} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-37+6\sqrt{30}}{17}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5}- 3 \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 2 \sqrt{5}- 3 \sqrt{6}\right) } \cdot \left( 2 \sqrt{5}- 3 \sqrt{6}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot- 3 \sqrt{6}\color{blue}{- 3 \sqrt{6}} \cdot 2 \sqrt{5}\color{blue}{- 3 \sqrt{6}} \cdot- 3 \sqrt{6} = \\ = 20- 6 \sqrt{30}- 6 \sqrt{30} + 54 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{5} + 3 \sqrt{6}\right) } \cdot \left( 2 \sqrt{5}- 3 \sqrt{6}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot- 3 \sqrt{6}+\color{blue}{ 3 \sqrt{6}} \cdot 2 \sqrt{5}+\color{blue}{ 3 \sqrt{6}} \cdot- 3 \sqrt{6} = \\ = 20- 6 \sqrt{30} + 6 \sqrt{30}-54 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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