\(\sqrt{\left(7+\sqrt{2}\right)^2}-\sqrt{\left(1-\sqrt{2}\right)^2}=\left|7+\sqrt{2}\right|-\left|1-\sqrt{2}\right|\)
\(=7+\sqrt{2}+1-\sqrt{2}=8\)
\(\sqrt{8-2\sqrt{7}}-\sqrt{8+2\sqrt{7}}=\sqrt{\left(\sqrt{7}-1\right)^2}-\sqrt{\left(\sqrt{7}+1\right)^2}\)
\(=\sqrt{7}-1-\left(\sqrt{7}+1\right)=-2\)
\(\sqrt{17+12\sqrt{2}}+\sqrt{17-12\sqrt{2}}=\sqrt{\left(3+2\sqrt{2}\right)^2}+\sqrt{\left(3-2\sqrt{2}\right)^2}\)
\(=3+2\sqrt{2}+3-2\sqrt{2}=6\)
\(\sqrt{a+2\sqrt{a-1}}+\sqrt{a-2\sqrt{a-1}}=\sqrt{\left(\sqrt{a-1}+1\right)^2}+\sqrt{\left(1-\sqrt{a-1}\right)^2}\)
\(=\left|\sqrt{a-1}+1\right|+\left|1-\sqrt{a-1}\right|=\sqrt{a-1}+1+1-\sqrt{a-1}=2\)
