a) \(\sqrt{10+4\sqrt{6}}-\sqrt{10-4\sqrt{6}}\)
\(=\sqrt{\left(\sqrt{6}\right)^2+2.\sqrt{6}.2+2^2}-\sqrt{\left(\sqrt{6}\right)^2-2.\sqrt{6}.2+2^2}\)
\(=\sqrt{\left(\sqrt{6}+2\right)^2}-\sqrt{\left(\sqrt{6}-2\right)^2}=\left|\sqrt{6}+2\right|-\left|\sqrt{6}-2\right|\)
\(=\sqrt{6}+2-\sqrt{6}+2=4\)
b) \(\sqrt{39-12\sqrt{3}}+\sqrt{39+12\sqrt{3}}\)
\(=\sqrt{6^2-2.6.\sqrt{3}+\left(\sqrt{3}\right)^2}+\sqrt{6^2+2.6.\sqrt{3}+\left(\sqrt{3}\right)^2}\)
\(=\sqrt{\left(6-\sqrt{3}\right)^2}+\sqrt{\left(6+\sqrt{3}\right)^2}=\left|6-\sqrt{3}\right|+\left|6+\sqrt{3}\right|\)
\(=6-\sqrt{3}+6+\sqrt{3}=12\)
c) \(\sqrt{31-12\sqrt{3}}-\sqrt{31+12\sqrt{3}}\)
\(=\sqrt{\left(3\sqrt{3}\right)^2-2.3\sqrt{3}.2+2^2}-\sqrt{\left(3\sqrt{3}\right)^2+2.3\sqrt{3}.2+2^2}\)
\(=\sqrt{\left(3\sqrt{3}-2\right)^2}-\sqrt{\left(3\sqrt{3}+2\right)^2}=\left|3\sqrt{3}-2\right|-\left|3\sqrt{3}+2\right|\)
\(=3\sqrt{3}-2-3\sqrt{3}-2=-4\)
d) \(\sqrt{21+12\sqrt{3}}+\sqrt{21-12\sqrt{3}}\)
\(=\sqrt{\left(2\sqrt{3}\right)^2+2.2\sqrt{3}.3+3^2}+\sqrt{\left(2\sqrt{3}\right)^2-2.2\sqrt{3}.3+3^2}\)
\(=\sqrt{\left(2\sqrt{3}+3\right)^2}+\sqrt{\left(2\sqrt{3}-3\right)^2}=\left|2\sqrt{3}+3\right|+\left|2\sqrt{3}-3\right|\)
\(=2\sqrt{3}+3+2\sqrt{3}-3=4\sqrt{3}\)
a) Ta có: \(\sqrt{10+4\sqrt{6}}-\sqrt{10-4\sqrt{6}}\)
\(=\sqrt{6}+2-\sqrt{6}+2\)
=4
b) Ta có: \(\sqrt{39-12\sqrt{3}}+\sqrt{39+12\sqrt{3}}\)
\(=6-\sqrt{3}+6+\sqrt{3}\)
=12
c) Ta có: \(\sqrt{31-12\sqrt{3}}-\sqrt{31+12\sqrt{3}}\)
\(=3\sqrt{3}-2-3\sqrt{3}-2\)
=-4
d) Ta có: \(\sqrt{21+12\sqrt{3}}+\sqrt{21-12\sqrt{3}}\)
\(=2\sqrt{3}+3+2\sqrt{3}-3\)
\(=4\sqrt{3}\)
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