1.
\(A=\sqrt{6+2\sqrt{5}}-\sqrt{6-2\sqrt{5}}\)
\(=\sqrt{5+2\sqrt{5}+1}-\sqrt{5-2\sqrt{5}+1}\)
\(=\sqrt{\left(\sqrt{5}+1\right)^2}-\sqrt{\left(\sqrt{5}-1\right)^2}\)
\(=\left|\sqrt{5}+1\right|-\left|\sqrt{5}-1\right|\)
\(=\sqrt{5}+1-\sqrt{5}+1=2\)
2.
\(B=\sqrt{7-2\sqrt{10}}+\sqrt{20}+\dfrac{1}{2}\sqrt{8}\)
\(=\sqrt{2+5-2\sqrt{2}.\sqrt{5}}+2\sqrt{5}+\sqrt{2}\)
\(=\sqrt{\left(\sqrt{5}-\sqrt{2}\right)^2}+2\sqrt{5}+\sqrt{2}\)
\(=\left|\sqrt{5}-\sqrt{2}\right|+2\sqrt{5}+\sqrt{2}\)
\(=\sqrt{5}-\sqrt{2}+2\sqrt{5}+\sqrt{2}=3\sqrt{5}\)
3.
\(C=\sqrt{17-12\sqrt{2}}+\sqrt{9+4\sqrt{2}}\)
\(=\sqrt{9+8-2.3.2\sqrt{2}}+\sqrt{8+1+2.2\sqrt{2}}\)
\(=\sqrt{\left(2\sqrt{2}-3\right)^2}+\sqrt{\left(2\sqrt{2}+1\right)^2}\)
\(=\left|2\sqrt{2}-3\right|+\left|2\sqrt{2}+1\right|\)
\(=3-2\sqrt{2}+2\sqrt{2}+1=4\)
4.
\(D=\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{3+1+2\sqrt{3}}-\sqrt{3+1-2\sqrt{3}}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left[\sqrt{\left(\sqrt{3}+1\right)^2}-\sqrt{\left(\sqrt{3}-1\right)^2}\right]\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{3}+1-\sqrt{3}+1\right)\)
\(=\dfrac{2}{\sqrt{2}}=\sqrt{2}\)
5.
\(E=\sqrt{6-4\sqrt{2}}+\sqrt{22-12\sqrt{2}}\)
\(=\sqrt{4+2-2.2.\sqrt{2}}+\sqrt{18+4-2.2.3\sqrt{2}}\)
\(=\sqrt{\left(2-\sqrt{2}\right)^2}+\sqrt{\left(3\sqrt{2}-2\right)^2}\)
\(=\left|2-\sqrt{2}\right|+\left|3\sqrt{2}-2\right|\)
\(=2-\sqrt{2}+3\sqrt{2}-2=2\sqrt{2}\)
1: Ta có: \(A=\sqrt{6+2\sqrt{5}}-\sqrt{6-2\sqrt{5}}\)
\(=\sqrt{5}+1-\sqrt{5}+1\)
=2
2: Ta có: \(B=\sqrt{7-2\sqrt{10}}+\sqrt{20}+\dfrac{1}{2}\sqrt{8}\)
\(=\sqrt{5}-\sqrt{2}+2\sqrt{5}+\sqrt{2}\)
\(=3\sqrt{5}\)
3: Ta có: \(C=\sqrt{17-12\sqrt{2}}+\sqrt{9+4\sqrt{2}}\)
\(=3-2\sqrt{2}+2\sqrt{2}+1\)
=4
4: Ta có: \(D=\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\)
\(=\dfrac{\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{3}+1-\sqrt{3}+1}{\sqrt{2}}=\sqrt{2}\)
5: Ta có: \(E=\sqrt{6-4\sqrt{2}}+\sqrt{22-12\sqrt{2}}\)
\(=2-\sqrt{2}+3\sqrt{2}-\sqrt{2}\)
\(=2+\sqrt{2}\)
