\(A=\frac{\sqrt{7-2\sqrt{10}}.\left(7+2\sqrt{10}\right)\left(74-22\sqrt{10}\right)}{\sqrt{125}-4\sqrt{50}+5\sqrt{20}+\sqrt{8}}\)

\(=\frac{\sqrt{\left(\sqrt{5}-\sqrt{2}\right)^2}.\left(78-6\sqrt{10}\right)}{5\sqrt{5}-20\sqrt{2}+10\sqrt{5}+2\sqrt{2}}\)

\(=\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(78-6\sqrt{10}\right)}{15\sqrt{5}-18\sqrt{2}}\)

\(=\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(26-2\sqrt{10}\right)}{5\sqrt{5}-6\sqrt{2}}\)

\(=\frac{30\sqrt{5}-36\sqrt{2}}{5\sqrt{5}-6\sqrt{2}}=6\)

\(2\left(\sqrt{n+1}-\sqrt{n}\right)=\frac{2}{\sqrt{n+1}+\sqrt{n}}< \frac{2}{2\sqrt{n}}=\frac{1}{\sqrt{n}}\)

\(2\left(\sqrt{n}-\sqrt{n-1}\right)=\frac{2}{\sqrt{n}+\sqrt{n-1}}>\frac{2}{2\sqrt{n}}=\frac{1}{\sqrt{n}}\)

\(\frac{1}{\sqrt{h+2\sqrt{h-1}}}+\frac{1}{\sqrt{h-2\sqrt{h-1}}}\)

\(=\frac{1}{\sqrt{h-1}+1}+\frac{1}{\sqrt{h-1}-1}\)

\(=\frac{2\sqrt{h-1}+1-1}{h-1+1}=\frac{2\sqrt{h-1}}{h}\)

ta có \(\frac{1}{1+a+ab}+\frac{1}{1+b+bc}+\frac{1}{1+c+ca}=\frac{abc}{abc+a+ab}+\frac{1}{1+b+bc}+\frac{abc}{abc+abc^2+ba^2c^2}\)

\(=\frac{abc}{a\left(bc+1+b\right)}+\frac{1}{1+b+bc}+\frac{abc}{ac\left(b+bc+abc\right)}\)

\(=\frac{bc}{1+b+bc}+\frac{1}{1+b+bc}+\frac{b}{1+b+bc}\)

\(=\frac{bc+1+b}{1+b+bc}=1\)

Aps dụng tính chất dãy tỉ số bằng nhau Ta có:

\(\frac{1+2+3}{x-1+y-2+z-3}=\frac{1+2+3}{x+y+z-1-2-3}=\frac{1+4+9}{x+2y+3z-\left(-4\right)}=\frac{ }{ }\)

=\(\frac{14}{56+4}=\frac{14}{60}=\frac{7}{30}\)

\(\Rightarrow\)\(\frac{1}{x-1}=\frac{7}{30}\)\(\Rightarrow\)x-1=\(\frac{30}{7}\)\(\Rightarrow\)x=\(\frac{37}{7}\)

\(\Rightarrow\)\(\frac{2}{y-2}=\frac{7}{30}\Rightarrow y-2=\frac{60}{7}\)\(\Rightarrow\)y=\(\frac{74}{7}\)

\(\Rightarrow\)\(\frac{3}{z-3}=\frac{7}{30}\Rightarrow z-3=\frac{90}{7}\)\(\Rightarrow\)x=\(\frac{111}{7}\)

Aps dụng tính chất dãy tỉ số bàng nhau, ta có:

\(\frac{1+2+3}{x-1+2y-2+z-3}=\frac{1+4+9}{x-1+2y-4+3z-9}\)=\(\frac{14}{x+2y+3z-1-2-3}=\frac{14}{56-1-2-3}=\frac{14}{50}=\frac{7}{25}\)

\(\Rightarrow\)\(\frac{1}{x-1}=\frac{7}{25}\Rightarrow x=\frac{32}{7}\)

\(\Rightarrow\)\(\frac{4}{2y-4}=\frac{7}{25}\Rightarrow y=\frac{64}{7}\)

\(\Rightarrow\)\(\frac{9}{3z-9}=\frac{7}{25}\Rightarrow z=\frac{96}{7}\)