Ta có: \(\sqrt{9-3\sqrt{5}}-\sqrt{9+3\sqrt{5}}\)
\(=\frac{\sqrt{2}\left(\sqrt{9-3\sqrt{5}}-\sqrt{9+3\sqrt{5}}\right)}{\sqrt{2}}\)
\(=\frac{\sqrt{18-6\sqrt{5}}-\sqrt{18+6\sqrt{5}}}{\sqrt{2}}\)
\(=\frac{\sqrt{15-2\cdot\sqrt{15}\cdot\sqrt{3}+3}-\sqrt{15+2\cdot\sqrt{15}\cdot\sqrt{3}+3}}{\sqrt{2}}\)
\(=\frac{\sqrt{\left(\sqrt{15}-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{15}+\sqrt{3}\right)^2}}{\sqrt{2}}\)
\(=\frac{\left|\sqrt{15}-\sqrt{3}\right|-\left|\sqrt{15}+\sqrt{3}\right|}{\sqrt{2}}\)
mà \(\left\{{}\begin{matrix}\sqrt{15}-\sqrt{3}>0\\\sqrt{15}+\sqrt{3}>0\end{matrix}\right.\)
nên \(\sqrt{9-3\sqrt{5}}-\sqrt{9+3\sqrt{5}}=\frac{\sqrt{15}-\sqrt{3}-\sqrt{15}-\sqrt{3}}{\sqrt{2}}=\frac{-2\sqrt{3}}{\sqrt{2}}=-\sqrt{6}\)
=\(\left(\sqrt{9-3\sqrt{5}}-\sqrt{9+3\sqrt{5}}\right)^2=9-3\sqrt{5}+9+3\sqrt{5}+2\sqrt{81-45}=18+12=30\)
