Question

Vreau sa ma ajutati, va rog

Answer 1

$\displaystyle x=\frac{\sqrt{3} +\sqrt{5} }{\sqrt{15} } -\frac{\sqrt{3}-\sqrt{7} }{\sqrt{21} } +\frac{\sqrt{5}-\sqrt{7} }{\sqrt{35} } =\\\\=\frac{\sqrt{7} \Big(\sqrt{3} +\sqrt{5}\Big) -\sqrt{5} \Big(\sqrt{3}-\sqrt{7} \Big)+\sqrt{3} \Big(\sqrt{5} -\sqrt{7} \Big) }{\sqrt{105} } =\\\\=\frac{\sqrt{21} +\sqrt{35}-\sqrt{15}+\sqrt{35} +\sqrt{15} -\sqrt{21} }{\sqrt{105} }=\frac{2\sqrt{35} }{\sqrt{105} } =\frac{2}{\sqrt{3} }$

$\displaystyle y=\sqrt{\frac{1}{4,1(6):0,5} } :0,4=\sqrt{\frac{1}{\cfrac{416-41}{90} :\cfrac{5}{10} } } :\frac{4}{10} =\sqrt{\frac{1}{\cfrac{375}{90} \cdot \cfrac{10}{5} } } \cdot \frac{10}{4} =\\\\=\sqrt{\frac{1}{\cfrac{375}{90} \cdot 2} } \cdot \frac{10}{4} =\sqrt{\frac{1}{\cfrac{375}{45} } }\cdot \frac{10}{4} =\sqrt{1 \cdot \frac{45}{375} } \cdot \frac{10}{4} =\sqrt{\frac{45}{375} }\cdot \frac{10}{4} =$

$\displaystyle =\frac{\sqrt{45} }{\sqrt{375} } \cdot \frac{10}{4} =\frac{3\sqrt{5} }{5\sqrt{15} }\cdot \frac{10}{4} =\frac{3}{5\sqrt{3} }\cdot \frac{5}{2} =\frac{15}{10\sqrt{3} }$

$\displaystyle m_g=\sqrt{x \cdot y}=\sqrt{\frac{2}{\sqrt{3} } \cdot \frac{15}{10\sqrt{3} } } =\sqrt{\frac{30}{10 \cdot 3} } =\sqrt{\frac{30}{30} }=\sqrt{1} =1\in\mathbb{N}$