\(12)\) \(\sqrt{10+2\sqrt{17-4\sqrt{9+4\sqrt{5}}}}\)
\(=\sqrt{10+2\sqrt{17-4\sqrt{\left(\sqrt{5}\right)^2+2\cdot\sqrt{5}\cdot2+2^2}}}\)
\(=\sqrt{10+2\sqrt{17-4\sqrt{\left(\sqrt{5}+2\right)^2}}}\)
\(=\sqrt{10+2\sqrt{17-4\left|\sqrt{5}+2\right|}}\)
\(=\sqrt{10+2\sqrt{17-4\sqrt{5}-8}}\)
\(=\sqrt{10+2\sqrt{9-4\sqrt{5}}}\)
\(=\sqrt{10+2\sqrt{\left(\sqrt{5}\right)^2-2\cdot\sqrt{5}\cdot2+2^2}}\)
\(=\sqrt{10+2\sqrt{\left(\sqrt{5}-2\right)^2}}\)
\(=\sqrt{10+2\left|\sqrt{5}-2\right|}\)
\(=\sqrt{10+2\sqrt{5}-4}\)
\(=\sqrt{6+2\sqrt{5}}\)
\(=\sqrt{\left(\sqrt{5}\right)^2+2\sqrt{5}+1}\)
\(=\sqrt{\left(\sqrt{5}+1\right)^2}\)
\(=\left|\sqrt{5}+1\right|=\sqrt{5}+1\)
\(13)\) \(\sqrt{6-2\sqrt{\sqrt{2}+\sqrt{12}+\sqrt{18-\sqrt{128}}}}\)
\(=\sqrt{6-2\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{\left(\sqrt{2}\right)^2-8\sqrt{2}+4^2}}}\)
\(=\sqrt{6-2\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{\left(\sqrt{2}-4\right)^2}}}\)
\(=\sqrt{6-2\sqrt{\sqrt{2}+2\sqrt{3}+\left|\sqrt{2}-4\right|}}\)
\(=\sqrt{6-2\sqrt{\sqrt{2}+2\sqrt{3}+4-\sqrt{2}}}\)
\(=\sqrt{6-2\sqrt{2\sqrt{3}+4}}\)
\(=\sqrt{6-2\sqrt{\left(\sqrt{3}\right)^2+2\sqrt{3}+1}}\)
\(=\sqrt{6-2\sqrt{\left(\sqrt{3}+1\right)^2}}\)
\(=\sqrt{6-2\left|\sqrt{3}+1\right|}\)
\(=\sqrt{6-2\sqrt{3}-2}\)
\(=\sqrt{4-2\sqrt{3}}\)
\(=\sqrt{\left(\sqrt{3}\right)^2-2\sqrt{3}+1}\)
\(=\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=\left|\sqrt{3}-1\right|=\sqrt{3}-1\)
#Toru
12:
\(=\sqrt{10+2\sqrt{17-4\left(\sqrt{5}+2\right)}}\)
\(=\sqrt{10+2\sqrt{9-4\sqrt{5}}}\)
\(=\sqrt{10+2\left(\sqrt{5}-2\right)}=\sqrt{6+2\sqrt{5}}=\sqrt{5}+1\)
13: \(=\sqrt{6-2\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{18-8\sqrt{2}}}}\)
\(=\sqrt{6-2\sqrt{\sqrt{2}+2\sqrt{3}+4-\sqrt{2}}}\)
\(=\sqrt{6-2\sqrt{4+2\sqrt{3}}}\)
\(=\sqrt{6-2\left(\sqrt{3}+1\right)}=\sqrt{4-2\sqrt{3}}=\sqrt{3}-1\)
