\(\sqrt{21-8\sqrt{5}}\)\(-\sqrt{21-4\sqrt{5}}\)
\(=\sqrt{16-2.4\sqrt{5}+5}\)\(-\sqrt{20-2\sqrt{20}+1}\)
\(=\sqrt{\left(4-\sqrt{5}\right)^2}\)\(-\sqrt{\left(\sqrt{20}-1\right)}\)
\(=4-\sqrt{5}-\left(\sqrt{20}-1\right)\)
\(=4-\sqrt{5}-\sqrt{20}+1\)
\(=5-\sqrt{5}-2\sqrt{5}\)
\(=5-3\sqrt{5}\)