Answer
$$\frac{3\sqrt{5}}{\sqrt{10}-2\sqrt{2}}=\frac{15\sqrt{2}+6\sqrt{10}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{5}}{\sqrt{10}-2\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{5}}{\sqrt{10}-2\sqrt{2}}\frac{\sqrt{10}+2\sqrt{2}}{\sqrt{10}+2\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{15\sqrt{2}+6\sqrt{10}}{10+4\sqrt{5}-4\sqrt{5}-8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15\sqrt{2}+6\sqrt{10}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{10} + 2 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 3 \sqrt{5} } \cdot \left( \sqrt{10} + 2 \sqrt{2}\right) = \color{blue}{ 3 \sqrt{5}} \cdot \sqrt{10}+\color{blue}{ 3 \sqrt{5}} \cdot 2 \sqrt{2} = \\ = 15 \sqrt{2} + 6 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{10}- 2 \sqrt{2}\right) } \cdot \left( \sqrt{10} + 2 \sqrt{2}\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot 2 \sqrt{2}\color{blue}{- 2 \sqrt{2}} \cdot \sqrt{10}\color{blue}{- 2 \sqrt{2}} \cdot 2 \sqrt{2} = \\ = 10 + 4 \sqrt{5}- 4 \sqrt{5}-8 $$ |
| ③ | Simplify numerator and denominator |
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