\(\sqrt{10+2\sqrt{17-4\sqrt{9+4\sqrt{5}}}}\)
\(=\sqrt{10+2\sqrt{17-4\sqrt{2^2+2\cdot2\sqrt{5}+\left(\sqrt{5}\right)^2}}}\)
\(=\sqrt{10+2\sqrt{17-4\sqrt{\left(2+\sqrt{5}\right)^2}}}\)
\(=\sqrt{10+2\sqrt{17-4\cdot\left|2+\sqrt{5}\right|}}\)
\(=\sqrt{10+2\sqrt{17-4\cdot\left(2+\sqrt{5}\right)}}\)
\(=\sqrt{10+2\sqrt{17-8-4\sqrt{5}}}\)
\(=\sqrt{10+2\sqrt{9-4\sqrt{5}}}\)
\(=\sqrt{10+2\sqrt{2^2-2\cdot2\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}}\)
\(=\sqrt{10+2\sqrt{\left(2-\sqrt{5}\right)^2}}\)
\(=\sqrt{10+2\cdot\left|2-\sqrt{5}\right|}\)
\(=\sqrt{10+2\cdot\left(-2+\sqrt{5}\right)}\)
\(=\sqrt{10+-4+2\sqrt{5}}\)
\(=\sqrt{6+2\sqrt{5}}\)
\(=\sqrt{\left(\sqrt{5}\right)^2+2\sqrt{5}\cdot1+1^2}\)
\(=\sqrt{\left(\sqrt{5}+1\right)^2}\)
\(=\left|\sqrt{5}+1\right|\)
\(=\sqrt{5}+1\)
\(=\sqrt{10+2\sqrt{17-4\sqrt{5+4\sqrt{5}+4}}}\)
\(=\sqrt{10+2\sqrt{17-4\sqrt{\left(\sqrt{5}+2\right)^2}}}\)
\(=\sqrt{10+2\sqrt{17-4\cdot\left|\sqrt{5}+2\right|}}\)
\(=\sqrt{10+2\sqrt{17-4\left(\sqrt{5}+2\right)}}\) (vì \(\sqrt{5}+2>0\))
\(=\sqrt{10+2\sqrt{17-4\sqrt{5}-8}}\)
\(=\sqrt{10+2\sqrt{9-4\sqrt{5}}}\\ =\sqrt{10+2\sqrt{5-4\sqrt{5}+4}}\\ =\sqrt{10+2\sqrt{\left(\sqrt{5}-2\right)^2}}\\ =\sqrt{10+2\cdot\left|\sqrt{5}-2\right|}\)
\(=\sqrt{10+2\cdot\left(\sqrt{5}-2\right)}\) (vì \(\sqrt{5}-2>0\))
\(=\sqrt{10+2\sqrt{5}-4}\\ =\sqrt{6+2\sqrt{5}}\\ =\sqrt{5+2\sqrt{5}+1}\\ =\sqrt{\left(\sqrt{5}+1\right)^2}\\ =\left|\sqrt{5}+1\right|\)
\(=\sqrt{5}+1\) (vì \(\sqrt{5}+1>0\))
