so sánh: \(4-\sqrt{2}\) và \(\sqrt{5}\)
\(\left(4-\sqrt{2}\right)^2=18-8\sqrt{2}>18-8\sqrt{2,25}=18-8.1,5=18-12=6>5=\sqrt{5}^2\Rightarrow4-\sqrt{2}>\sqrt{5}\left(vì:\left\{{}\begin{matrix}4-\sqrt{2}>0\\\sqrt{5}>0\end{matrix}\right.\right)\Rightarrow7+4-\sqrt{2}>7+\sqrt{5}\Rightarrow11-\sqrt{2}>7+\sqrt{5}\)
\(b,2006^2-2005.2007=2006^2-\left(2006-1\right)\left(2006+1\right)=2006^2-2006^2+1=1\Rightarrow2006^2>2005.2007\left(1\right)\)
\(\left(2\sqrt{2006}\right)^2=4.2006=8024;\left(\sqrt{2005}+\sqrt{2007}\right)^2=2005+2007+2\sqrt{2005.2007}=4012+2\sqrt{2005.2007}=4012+2\sqrt{2006.2006}\left(vì\left(1\right)\right)=8024=\left(2\sqrt{2006}\right)^2\)
\(\Rightarrow\sqrt{2005}+\sqrt{2007}< 2\sqrt{2006}\left(vì:\left\{{}\begin{matrix}\sqrt{2005}+\sqrt{2007}>0\\2\sqrt{2006}>0\end{matrix}\right.\right)\)
\(c,\left(\sqrt{10}+\sqrt{13}\right)^2=23+2\sqrt{130}>23+2\sqrt{121}\left(130>121\right)=23+2.11=45>4.11=\left(2\sqrt{11}\right)^2\Rightarrow\sqrt{10}+\sqrt{13}>2\sqrt{11}\left(vì\left\{{}\begin{matrix}\sqrt{10}+\sqrt{13}>0\\2\sqrt{11}>0\end{matrix}\right.\right)\)
\(d,\left(\sqrt{5}+\sqrt{7}\right)^2=12+2\sqrt{35}< 12+2\sqrt{36}=12+12=24< 15+6\sqrt{6}=\left(3+\sqrt{6}\right)^2\Rightarrow\sqrt{5}+\sqrt{7}< 3+\sqrt{6}\left(vì:\left\{{}\begin{matrix}\sqrt{5}+\sqrt{7}>0\\3+\sqrt{6}>0\end{matrix}\right.\right)\)
