Answer
$$\frac{3\sqrt{5}+\sqrt{5}\cdot\sqrt{3}+\sqrt{3}\cdot(\sqrt{5}+\sqrt{3})}{\sqrt{5}-\sqrt{3}\cdot\sqrt{5}+\sqrt{3}}=\frac{3\sqrt{5}+2\sqrt{15}+3}{\sqrt{5}-\sqrt{3}\cdot\sqrt{5}+\sqrt{3}}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{5}+\sqrt{5}\cdot\sqrt{3}+\sqrt{3}\cdot(\sqrt{5}+\sqrt{3})}{\sqrt{5}-\sqrt{3}\cdot\sqrt{5}+\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\frac{3\sqrt{5}+\sqrt{15}+\sqrt{15}+3}{1}}{\sqrt{5}-\sqrt{3}\cdot\sqrt{5}+\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3\sqrt{5}+2\sqrt{15}+3}{\sqrt{5}-\sqrt{3}\cdot\sqrt{5}+\sqrt{3}}\end{aligned} $$ | |
| ① | $$ 3\sqrt{5}+\sqrt{5}\cdot\sqrt{3}+\sqrt{3}\cdot(\sqrt{5}+\sqrt{3})
= 3\sqrt{5}+\sqrt{15} \cdot \color{blue}{\frac{ 1 }{ 1}} + \sqrt{15}+3 \cdot \color{blue}{\frac{ 1 }{ 1}}
= \frac{3\sqrt{5}+\sqrt{15}+\sqrt{15}+3}{1} $$ |
| ② | Remove 1 from denominator. |
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