Question
Answer
$$\dfrac{5\cdot(2-\sqrt{3})}{2\cdot(3+\sqrt{2})}=\dfrac{60-20\sqrt{2}-30\sqrt{3}+10\sqrt{6}}{28}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\cdot(2-\sqrt{3})}{2\cdot(3+\sqrt{2})}& \xlongequal{ }(10-5\sqrt{3})\cdot\frac{1}{6+2\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{10-5\sqrt{3}}{6+2\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10-5\sqrt{3}}{6+2\sqrt{2}}\frac{6-2\sqrt{2}}{6-2\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{60-20\sqrt{2}-30\sqrt{3}+10\sqrt{6}}{36-12\sqrt{2}+12\sqrt{2}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{60-20\sqrt{2}-30\sqrt{3}+10\sqrt{6}}{28}\end{aligned} $$ | |
| ① | $$ \color{blue}{ \left( 10- 5 \sqrt{3}\right) } \cdot 1 = \color{blue}{10} \cdot1\color{blue}{- 5 \sqrt{3}} \cdot1 = \\ = 10- 5 \sqrt{3} $$$$ \color{blue}{ 1 } \cdot \left( 6 + 2 \sqrt{2}\right) = \color{blue}{1} \cdot6+\color{blue}{1} \cdot 2 \sqrt{2} = \\ = 6 + 2 \sqrt{2} $$ |
| ② | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6- 2 \sqrt{2}} $$. |
| ③ | Multiply in a numerator. $$ \color{blue}{ \left( 10- 5 \sqrt{3}\right) } \cdot \left( 6- 2 \sqrt{2}\right) = \color{blue}{10} \cdot6+\color{blue}{10} \cdot- 2 \sqrt{2}\color{blue}{- 5 \sqrt{3}} \cdot6\color{blue}{- 5 \sqrt{3}} \cdot- 2 \sqrt{2} = \\ = 60- 20 \sqrt{2}- 30 \sqrt{3} + 10 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 6 + 2 \sqrt{2}\right) } \cdot \left( 6- 2 \sqrt{2}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot- 2 \sqrt{2}+\color{blue}{ 2 \sqrt{2}} \cdot6+\color{blue}{ 2 \sqrt{2}} \cdot- 2 \sqrt{2} = \\ = 36- 12 \sqrt{2} + 12 \sqrt{2}-8 $$ |
| ④ | Simplify numerator and denominator |
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