\(\sqrt{41-12\sqrt{5}}-\sqrt{41+12\sqrt{5}}\)
\(=\sqrt{36-12\sqrt{5}+5}-\sqrt{36+12\sqrt{5}+5}\)
\(=\sqrt{6^2-2\cdot6\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}-\sqrt{6^2+2\cdot6\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}\)
\(=\sqrt{\left(6-\sqrt{5}\right)^2}-\sqrt{\left(6+\sqrt{5}\right)^2}\)
\(=\left|6-\sqrt{5}\right|-\left|6+\sqrt{5}\right|\)
\(=6-\sqrt{5}-6-\sqrt{5}\)
\(=-2\sqrt{5}\)
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