\(\frac{\sqrt{3+\sqrt{15}}}{\sqrt{2}}=\sqrt{\frac{3+\sqrt{15}}{2}}\)
\(=\sqrt{\frac{\sqrt{3}\left(\sqrt{3}+\sqrt{5}\right)}{5-3}}\)
\(=\sqrt{\frac{\sqrt{3}\left(\sqrt{3}+\sqrt{5}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}}\)
\(=\sqrt{\frac{\sqrt{3}}{\sqrt{5}-\sqrt{3}}}\)
a) \(\left(\sqrt{12}-\sqrt{27}+\sqrt{3}\right):\sqrt{3}\)
\(=\left(2\sqrt{3}-3\sqrt{3}+\sqrt{3}\right):\sqrt{3}\)
\(=\sqrt{3}\left(2-3+1\right):\sqrt{3}\)
\(=0:\sqrt{3}=0\)
b) \(\left(5\sqrt{3}+3\sqrt{5}\right):\sqrt{15}\)
\(=\frac{5\sqrt{3}}{\sqrt{15}}+\frac{3\sqrt{5}}{\sqrt{15}}\)
\(=\frac{5\sqrt{3}}{\sqrt{3}\cdot\sqrt{5}}+\frac{3\sqrt{5}}{\sqrt{3}\cdot\sqrt{5}}\)
\(=\sqrt{5}+\sqrt{3}\)
Nguyễn Thị Thu Sương b) có cách khác nữa nè ( cơ bản là giống :v )
\(\left(5\sqrt{3}+3\sqrt{5}\right):\sqrt{15}\)
\(=\left(\sqrt{5}\cdot\sqrt{5}\cdot\sqrt{3}+\sqrt{3}\cdot\sqrt{3}\cdot\sqrt{5}\right):\left(\sqrt{3}\cdot\sqrt{5}\right)\)
\(=\sqrt{3}\cdot\sqrt{5}\cdot\left(\sqrt{5}+\sqrt{3}\right):\left(\sqrt{3}\cdot\sqrt{5}\right)\)
\(=\sqrt{5}+\sqrt{3}\)
