m) \(\sqrt{82+12\sqrt{42}}=\sqrt{\left(3\sqrt{6}\right)^2+2.3\sqrt{6}.2\sqrt{7}+\left(2\sqrt{7}\right)^2}\)
\(=\sqrt{\left(3\sqrt{6}+2\sqrt{7}\right)^2}=3\sqrt{6}+2\sqrt{7}\)
o) \(\sqrt{14-6\sqrt{5}}+\sqrt{14-6\sqrt{5}}\)
\(=\sqrt{3^2-2.3.\sqrt{5}+\left(\sqrt{5}\right)^2}+\sqrt{3^2+2.3.\sqrt{5}+\left(\sqrt{5}\right)^2}\)
\(=\sqrt{\left(3-\sqrt{5}\right)^2}+\sqrt{\left(3+\sqrt{5}\right)^2}=3-\sqrt{5}+3+\sqrt{5}=6\)
v) \(\sqrt{18+4\sqrt{6}+8\sqrt{3}+4\sqrt{2}}\)
\(=\sqrt{2^2+\left(\sqrt{2}\right)^2+\left(2\sqrt{3}\right)^2+2.2\sqrt{3}.\sqrt{2}+2.2\sqrt{3}.2+2.2.\sqrt{2}}\)
\(=\sqrt{\left(2+\sqrt{2}+2\sqrt{3}\right)^2}=2+\sqrt{2}+2\sqrt{3}\)
w) \(\sqrt{49-5\sqrt{96}}+\sqrt{49+5\sqrt{96}}=\sqrt{\dfrac{98-10\sqrt{96}}{2}}+\sqrt{\dfrac{98+10\sqrt{96}}{2}}\)
\(=\sqrt{\dfrac{\left(5\sqrt{2}\right)^2-2.5\sqrt{2}.4\sqrt{3}+\left(4\sqrt{3}\right)^2}{2}}+\sqrt{\dfrac{\left(5\sqrt{2}\right)^2+2.5\sqrt{2}.4\sqrt{3}+\left(4\sqrt{3}\right)^2}{2}}\)
\(=\sqrt{\dfrac{\left(5\sqrt{2}-4\sqrt{3}\right)^2}{2}}+\sqrt{\dfrac{\left(5\sqrt{2}+4\sqrt{3}\right)^2}{2}}\)
\(=\dfrac{5\sqrt{2}-4\sqrt{3}}{\sqrt{2}}+\dfrac{5\sqrt{2}+4\sqrt{3}}{\sqrt{2}}=\dfrac{10\sqrt{2}}{\sqrt{2}}=10\)
m) \(\sqrt{82+12\sqrt{42}}=3\sqrt{6}+2\sqrt{7}\)
o) \(\sqrt{14-6\sqrt{5}}+\sqrt{14+6\sqrt{5}}=3-\sqrt{5}+3+\sqrt{5}=6\)
r) \(\sqrt{11-6\sqrt{2}}+\sqrt{3-2\sqrt{2}}=3-\sqrt{2}+\sqrt{2}-1=2\)
