\(a,\sqrt{\sqrt{17+12\sqrt{2}}}\)
\(=\sqrt{\sqrt{8+12\sqrt{2}+9}}\)
\(=\sqrt{\sqrt{\left[2\sqrt{2}+3\right]^2}}\)
\(=\sqrt{2\sqrt{2}+3}\)
\(=\sqrt{1+2\sqrt{2}+2}\)
\(=\sqrt{\left[1+\sqrt{2}\right]^2}\)
\(=1+\sqrt{2}\)
\(b,\sqrt{4+2\sqrt{3}}-\sqrt{21-12\sqrt{3}}\)
\(=\sqrt{3+2\sqrt{3}+1}-\sqrt{12-12\sqrt{3}+9}\)
\(=\sqrt{\left[1+\sqrt{3}\right]^2}-\sqrt{\left[2\sqrt{3}-3\right]^2}\)
\(=\left(1+\sqrt{3}\right)-\left(2\sqrt{3}-3\right)\)
\(=1+\sqrt{3}-2\sqrt{3}+3\)
\(=4-\sqrt{3}\)
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