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\(A=2\sqrt{1}+2\sqrt{3}+...+2\sqrt{21}\)
\(A=2.\left(\sqrt{1}+\sqrt{3}+...+\sqrt{21}\right)\)
\(B=2\sqrt{2}+2\sqrt{4}+....2\sqrt{22}\)
\(B=2.\left(\sqrt{2}+\sqrt{4}+...+\sqrt{22}\right)\)
Có \(\sqrt{1}+\sqrt{3}+...+\sqrt{21}\) Có 11 số hạng.
\(\sqrt{2}+\sqrt{4}+...+\sqrt{22}\) Có 11 số hạng.
Mà \(\hept{\begin{cases}\sqrt{1}< \sqrt{2}\\....\\\sqrt{21}< \sqrt{22}\end{cases}}\)
=> \(2.\left(\sqrt{1}+\sqrt{3}+...+\sqrt{21}\right)< 2.\left(\sqrt{2}+\sqrt{4}+...+\sqrt{22}\right)\)
\(\Rightarrow A< B\)
a) \(\frac{\sqrt{110}+\sqrt{70}}{\sqrt{22}+\sqrt{14}}=\frac{\left(\sqrt{11}+\sqrt{7}\right)\sqrt{10}}{\left(\sqrt{11}+\sqrt{7}\right)\sqrt{2}}=\sqrt{5}\)
b) \(\frac{\sqrt{42}-6}{\sqrt{21}-\sqrt{18}}=\frac{\sqrt{42}-\sqrt{36}}{\sqrt{21}-\sqrt{18}}\)
\(=\frac{\left(\sqrt{7}-\sqrt{6}\right)\sqrt{6}}{\left(\sqrt{7}-\sqrt{6}\right)\sqrt{3}}=\sqrt{2}\)
c) \(\frac{\left(a-b\right)\sqrt{a^2-b^2}}{\left(a-b\right)^2}\)
\(=\frac{\sqrt{\left(a-b\right)\left(a+b\right)}}{a-b}\)
Bài 2 :
a) \(A=\sqrt{8+2\sqrt{7}}-\sqrt{7}=\sqrt{7+2\sqrt{7}+1}-\sqrt{7}\)
\(=\sqrt{\left(\sqrt{7}+1\right)^2}-\sqrt{7}=\left|\sqrt{7}+1\right|-\sqrt{7}=\sqrt{7}+1-\sqrt{7}=1\)
b) \(B=\sqrt{7+4\sqrt{3}}-2\sqrt{3}=\sqrt{4+4\sqrt{3}+3}-2\sqrt{3}\)
\(=\sqrt{\left(2+\sqrt{3}\right)^2}-2\sqrt{3}=\left|2+\sqrt{3}\right|-2\sqrt{3}\)
\(=2+\sqrt{3}-2\sqrt{3}=2-\sqrt{3}\)
c) \(C=\sqrt{14-2\sqrt{13}}+\sqrt{14+2\sqrt{13}}\)
\(=\sqrt{13-2\sqrt{13}+1}+\sqrt{13+2\sqrt{13}+1}\)
\(=\sqrt{\left(\sqrt{13}-1\right)^2}+\sqrt{\left(\sqrt{13}+1\right)^2}\)
\(=\left|\sqrt{13}-1\right|+\left|\sqrt{13}+1\right|\)
\(=\sqrt{13}-1+\sqrt{13}+1=2\sqrt{13}\)
d) \(D=\sqrt{22-2\sqrt{21}}+\sqrt{22+2\sqrt{21}}\)
\(=\sqrt{21-2\sqrt{21}+1}+\sqrt{21+2\sqrt{21}+1}\)
\(=\sqrt{\left(\sqrt{21}-1\right)^2}+\sqrt{\left(\sqrt{21}+1\right)^2}\)
\(=\left|\sqrt{21}-1\right|+\left|\sqrt{21}+1\right|\)
\(=\sqrt{21}-1+\sqrt{21}+1=2\sqrt{21}\)
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