\(P=\sqrt{11+6\sqrt{2}}-\sqrt{11-6\sqrt{2}}\\ =\sqrt{9+2+6\sqrt{2}}-\sqrt{9+2-6\sqrt{2}}\\ =\sqrt{\left(3+\sqrt{2}\right)^2}-\sqrt{\left(3-\sqrt{2}\right)^2}\\ =3+\sqrt{2}-3+\sqrt{2}\\ =2\sqrt{2}\)
\(Q=\sqrt{17+12\sqrt{2}}+\sqrt{17-12\sqrt{2}}\\ =\sqrt{9+8+6\sqrt{8}}+\sqrt{9+8-6\sqrt{8}}\\ =\sqrt{\left(3+\sqrt{8}\right)^2}+\sqrt{\left(3-\sqrt{8}\right)^2}\\ =3+\sqrt{8}+3-\sqrt{8}\\ =6\)
a) \(P=\sqrt{11+6\sqrt{2}}-\sqrt{11-6\sqrt{2}}=\sqrt{9+2.3.\sqrt{2}+2}-\sqrt{9-2.3.\sqrt{2}+2}=\sqrt{\left(3+\sqrt{2}\right)^2}-\sqrt{\left(3-\sqrt{2}\right)^2}=\left|3+\sqrt{2}\right|-\left|3-\sqrt{2}\right|=3+\sqrt{2}-3+\sqrt{2}=2\sqrt{2}\)
b) \(Q=\sqrt{17+12\sqrt{2}}+\sqrt{17-12\sqrt{2}}=\sqrt{9+2.3.2\sqrt{2}+8}+\sqrt{9-2.3.2\sqrt{2}+8}=\sqrt{\left(3+2\sqrt{2}\right)^2}+\sqrt{\left(3-2\sqrt{2}\right)^2}=\left|3+2\sqrt{2}\right|+\left|3-2\sqrt{2}\right|=3+2\sqrt{2}+3-2\sqrt{2}=6\)
