\(\text{a},=6\sqrt{2}-7\sqrt{2}+9\sqrt{2}=8\sqrt{2}\\ b,=5\sqrt{5}-4\sqrt{5}-12\sqrt{5}+12\sqrt{5}=\sqrt{5}\\ c,=\sqrt{17-3\sqrt{16.2}}+\sqrt{17+3\sqrt{16.2}}=\sqrt{17-12\sqrt{2}}+\sqrt{17+12\sqrt{2}}=\sqrt{\left(\sqrt{6}-\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{6}+\sqrt{2}\right)^2}\)
=I\(\sqrt{6}-\sqrt{2}\)I+I\(\sqrt{6}+\sqrt{2}\)I=\(\sqrt{6}-\sqrt{2}+\sqrt{6}+\sqrt{2}=2\sqrt{6}\)
d) \(\sqrt{49-5\sqrt{96}}-\sqrt{49+5\sqrt{96}}\)
\(=\sqrt{49-5\sqrt{16.6}}-\sqrt{49+5\sqrt{16.6}}\)
\(=\sqrt{49-20\sqrt{6}}-\sqrt{49+20\sqrt{6}}\)
\(=\sqrt{\left(5-2\sqrt{6}\right)^2}-\sqrt{\left(5+2\sqrt{6}\right)^2}\)
\(=\left|5-2\sqrt{6}\right|-\left|5+2\sqrt{6}\right|\)
\(=5-2\sqrt{6}-5-2\sqrt{6}=-4\sqrt{6}\).
e) \(2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}\)
\(=2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{16.3}}}}\)
\(=2\sqrt{3+\sqrt{5-\sqrt{13+4\sqrt{3}}}}\)
\(=2\sqrt{3+\sqrt{5-\sqrt{\left(1+2\sqrt{3}\right)^2}}}\)
\(=2\sqrt{3+\sqrt{5-\left|1+2\sqrt{3}\right|}}\)
\(=2\sqrt{3+\sqrt{5-1-2\sqrt{3}}}\)
\(=2\sqrt{3+\sqrt{4-2\sqrt{3}}}\)
\(=2\sqrt{3+\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(=2\sqrt{3+\left|\sqrt{3}-1\right|}\)
\(=2\sqrt{2+\sqrt{3}}\)
\(=\sqrt{2}.\sqrt{2}.\sqrt{2+\sqrt{3}}\)
\(=\sqrt{2}.\sqrt{4+2\sqrt{3}}\)
\(=\sqrt{2}.\sqrt{\left(\sqrt{3}+1\right)^2}\)
\(=\sqrt{2}.\left|\sqrt{3}+1\right|=\sqrt{6}+\sqrt{2}\)
g) \(\sqrt{4+\sqrt{8}}.\sqrt{2+\sqrt{2+\sqrt{2}}}.\sqrt{2-\sqrt{2+\sqrt{2}}}\)
\(=\sqrt{4+\sqrt{8}}.\sqrt{\left(2+\sqrt{2+\sqrt{2}}\right)\left(2-\sqrt{2+\sqrt{2}}\right)}\)
\(=\sqrt{4+\sqrt{8}}.\sqrt{4-\left(2+\sqrt{2}\right)}\)
\(=\sqrt{4+\sqrt{8}}.\sqrt{2-\sqrt{2}}\)
\(=\sqrt{2\left(2+\sqrt{2}\right)}.\sqrt{2-\sqrt{2}}\)
\(=\sqrt{2\left(2+\sqrt{2}\right)\left(2-\sqrt{2}\right)}\)
\(=\sqrt{2\left(4-2\right)}=2\)
