Question

Tính :

\(\sqrt{29-2\sqrt{180}}-\sqrt{9+4\sqrt{5}}\)

Answer 1

\(\sqrt{29-2\sqrt{180}}-\sqrt{9+4\sqrt{5}}\)

\(=\sqrt{29-2\sqrt{6^2.5}}-\sqrt{9+4\sqrt{5}}\)

\(=\sqrt{29-12\sqrt{5}}-\sqrt{9+4\sqrt{5}}\)

\(=\sqrt{\left(2\sqrt{5}\right)^2-2.2\sqrt{5}.3+3^2}-\sqrt{\left(\sqrt{5}\right)^2+2.\sqrt{5}.2+2^2}\)

\(=\sqrt{\left(2\sqrt{5}-3\right)^2}-\sqrt{\left(\sqrt{5}+2\right)^2}\)

\(=\left|2\sqrt{5}-3\right|-\left|\sqrt{5}+2\right|\)

\(=2\sqrt{5}-3-\left(\sqrt{5}+2\right)\) (vì \(2\sqrt{5}-3>0,\sqrt{5}+2>0\))

\(=2\sqrt{5}-3-\sqrt{5}-2\)

\(=\sqrt{5}-5\)