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Ta có : \(\sqrt{3}-\sqrt{2}=\dfrac{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}{\sqrt{3}+\sqrt{2}}=\dfrac{1}{\sqrt{3}+\sqrt{2}}>\dfrac{1}{\sqrt{4}+\sqrt{3}}=\sqrt{4}-\sqrt{3}\Rightarrow\sqrt{3}-\sqrt{2}>\sqrt{4}-\sqrt{3}\Rightarrow2\sqrt{3}>\sqrt{4}+\sqrt{2}\)
Làm tương tự : \(2\sqrt{5}>\sqrt{4}+\sqrt{6};2\sqrt{7}>\sqrt{6}+\sqrt{8},...,2\sqrt{19}>\sqrt{18}+\sqrt{20}\)
Cộng từng BĐT trên , ta được :
\(2\sqrt{3}+2\sqrt{5}+...+2\sqrt{19}>\sqrt{4}+\sqrt{2}+\sqrt{4}+\sqrt{6}+...+\sqrt{18}+\sqrt{20}=2\sqrt{4}+2\sqrt{6}+...+2\sqrt{18}+\sqrt{20}+\sqrt{2}\)
\(\Leftrightarrow A-2\sqrt{1}>B-\sqrt{2}\)
\(\Leftrightarrow A-B>2-\sqrt{2}>0\Rightarrow A>B\)
a) \(\sqrt{6+2\sqrt{5}}+\sqrt{6-2\sqrt{5}}=\sqrt{1+2\sqrt{5}+\left(\sqrt{5}\right)^2}+\sqrt{1-2\sqrt{5}+\left(\sqrt{5}\right)^2}\)\(=\sqrt{\left(1+\sqrt{5}\right)^2}+\sqrt{\left(1-\sqrt{5}\right)^2}=1+\sqrt{5}-\left(1-\sqrt{5}\right)=1+\sqrt{5}-1+\sqrt{5}=2\sqrt{5}\)
a) \(\sqrt{6+2\sqrt{5}}+\sqrt{6-2\sqrt{5}}\)
\(=\sqrt{\left(\sqrt{5}+1\right)^2}+\sqrt{\left(\sqrt{5}-1\right)^2}\)
\(=\sqrt{5}+1+\sqrt{5}-1=2\sqrt{5}\)
b) \(\sqrt{13+4\sqrt{10}}+\sqrt{13-4\sqrt{10}}\)
\(=\sqrt{\left(2\sqrt{2}+\sqrt{5}\right)^2}+\sqrt{\left(2\sqrt{2}-\sqrt{5}\right)^2}\)
\(=2\sqrt{2}+\sqrt{5}+2\sqrt{2}-\sqrt{5}=4\sqrt{2}\)
c) \(\sqrt{8\sqrt{3}}-2\sqrt{25\sqrt{12}}+4\sqrt{\sqrt{192}}\)
\(=2\sqrt{2\sqrt{3}}-10\sqrt{2\sqrt{3}}+8\sqrt{2\sqrt{3}}=0\)
\(a,\sqrt{12}+2\sqrt{27}+3\sqrt{75}-9\sqrt{48}=2\sqrt{3}+6\sqrt{3}+15\sqrt{3}-36\sqrt{3}\)
\(=-13\sqrt{3}\)
\(b,2\sqrt{3}.\left(\sqrt{27}+2\sqrt{48}-\sqrt{75}\right)=2\sqrt{3}\left(3\sqrt{3}+8\sqrt{3}-5\sqrt{3}\right)\)
\(=2\sqrt{3}.6\sqrt{3}=36\)
\(c,\left(2\sqrt{2}-\sqrt{3}\right)^2=8-4\sqrt{6}+3\)
\(=11-4\sqrt{6}\)
\(d,\left(1+\sqrt{3}-\sqrt{2}\right)\left(1+\sqrt{3}+\sqrt{2}\right)=1+2\sqrt{3}+3-2\)
\(=2+2\sqrt{3}\)
Xét hiệu :
\(A-B=2\left(\sqrt{1}-\sqrt{2}\right)+2.\left(\sqrt{3}-\sqrt{4}\right)+...+2\left(\sqrt{19}-\sqrt{20}\right)\)
Mà: \(\sqrt{1}<\sqrt{2};\sqrt{3}<\sqrt{4};...;\sqrt{19}<\sqrt{20}\)
nên \(\sqrt{1}-\sqrt{2}<0;\sqrt{3}-\sqrt{4}<0;...;\sqrt{19}-\sqrt{20}<0\)
=> A - B < 0 => A < B