Question
Answer
$$\dfrac{3-2\sqrt{5}}{3\sqrt{8}+\sqrt{2}}=\dfrac{3\sqrt{2}-2\sqrt{10}}{14}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3-2\sqrt{5}}{3\sqrt{8}+\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3-2\sqrt{5}}{3\sqrt{8}+\sqrt{2}}\frac{3\sqrt{8}-\sqrt{2}}{3\sqrt{8}-\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{18\sqrt{2}-3\sqrt{2}-12\sqrt{10}+2\sqrt{10}}{72-12+12-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15\sqrt{2}-10\sqrt{10}}{70} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3\sqrt{2}-2\sqrt{10}}{14}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{8}- \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3- 2 \sqrt{5}\right) } \cdot \left( 3 \sqrt{8}- \sqrt{2}\right) = \color{blue}{3} \cdot 3 \sqrt{8}+\color{blue}{3} \cdot- \sqrt{2}\color{blue}{- 2 \sqrt{5}} \cdot 3 \sqrt{8}\color{blue}{- 2 \sqrt{5}} \cdot- \sqrt{2} = \\ = 18 \sqrt{2}- 3 \sqrt{2}- 12 \sqrt{10} + 2 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{8} + \sqrt{2}\right) } \cdot \left( 3 \sqrt{8}- \sqrt{2}\right) = \color{blue}{ 3 \sqrt{8}} \cdot 3 \sqrt{8}+\color{blue}{ 3 \sqrt{8}} \cdot- \sqrt{2}+\color{blue}{ \sqrt{2}} \cdot 3 \sqrt{8}+\color{blue}{ \sqrt{2}} \cdot- \sqrt{2} = \\ = 72-12 + 12-2 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
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