Question
Answer
$$\dfrac{\sqrt{10}}{6-2\sqrt{5}}=\dfrac{3\sqrt{10}+5\sqrt{2}}{8}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{10}}{6-2\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{10}}{6-2\sqrt{5}}\frac{6+2\sqrt{5}}{6+2\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6\sqrt{10}+10\sqrt{2}}{36+12\sqrt{5}-12\sqrt{5}-20} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6\sqrt{10}+10\sqrt{2}}{16} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3\sqrt{10}+5\sqrt{2}}{8}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 6 + 2 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{10} } \cdot \left( 6 + 2 \sqrt{5}\right) = \color{blue}{ \sqrt{10}} \cdot6+\color{blue}{ \sqrt{10}} \cdot 2 \sqrt{5} = \\ = 6 \sqrt{10} + 10 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( 6- 2 \sqrt{5}\right) } \cdot \left( 6 + 2 \sqrt{5}\right) = \color{blue}{6} \cdot6+\color{blue}{6} \cdot 2 \sqrt{5}\color{blue}{- 2 \sqrt{5}} \cdot6\color{blue}{- 2 \sqrt{5}} \cdot 2 \sqrt{5} = \\ = 36 + 12 \sqrt{5}- 12 \sqrt{5}-20 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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