a: \(\sqrt{6-3\sqrt3}+\sqrt{2-\sqrt3}\)
\(=\frac{\sqrt{12-6\sqrt3}+\sqrt{4-2\sqrt3}}{\sqrt2}\)
\(=\frac{\sqrt{\left(3-\sqrt3\right)^2}+\sqrt{\left(\sqrt3-1\right)^2}}{\sqrt2}\)
\(=\frac{\left(3-\sqrt3\right)^{}+\left(\sqrt3-1\right)}{\sqrt2}=\frac{2}{\sqrt2}=\sqrt2\)
b: \(\sqrt{9+\sqrt{17}}-\sqrt{9-\sqrt{17}}-\sqrt2\)
\(=\frac{\sqrt{18+2\sqrt{17}}-\sqrt{18-2\sqrt{17}}}{\sqrt2}-\sqrt2\)
\(=\frac{\sqrt{\left(\sqrt{17}+1\right)^2}-\sqrt{\left(\sqrt{17}-1\right)^2}}{\sqrt2}-\sqrt2\)
\(=\frac{\left(\sqrt{17}+1\right)-\left(\sqrt{17}-1\right)}{\sqrt2}-\sqrt2=\frac{2}{\sqrt2}-\sqrt2=0\)
c: \(\sqrt{24-3\sqrt{15}}-\sqrt{36-9\sqrt{15}}\)
\(=\sqrt{3\left(8-\sqrt{15}\right)}-\sqrt{9\left(4-\sqrt{15}\right)}\)
\(=\frac{\sqrt{3\left(16-2\sqrt{15}\right)}-\sqrt{9\left(8-2\sqrt{15}\right)}}{\sqrt2}\)
\(=\frac{\sqrt{3\left(\sqrt{15}-1\right)^2}-3\sqrt{\left(\sqrt5-\sqrt3\right)^2}}{\sqrt2}\)
\(=\frac{\sqrt3\cdot\left(\sqrt{15}-1\right)-3\left(\sqrt5-\sqrt3\right)}{\sqrt2}=\frac{3\sqrt5-\sqrt3-3\sqrt5+3\sqrt3}{\sqrt2}=\frac{2\sqrt3}{\sqrt2}=\sqrt6\)
d: \(\sqrt2\cdot\sqrt{2-\sqrt3}\left(\sqrt3+1\right)\)
\(=\sqrt{4-2\sqrt3}\left(\sqrt3+1\right)\)
\(=\sqrt{\left(\sqrt3-1\right)^2}\cdot\left(\sqrt3+1\right)=\left(\sqrt3-1\right)\left(\sqrt3+1\right)\)
=3-1
=2
e: \(\frac{\sqrt{4+\sqrt7}}{\sqrt2}=\frac{\sqrt{8+2\sqrt7}}{2}=\frac{\sqrt{\left(\sqrt7+1\right)^2}}{2}=\frac{\sqrt7+1}{2}\)
f: \(\frac{2+\sqrt3}{3+\sqrt3}=\frac{\left(2+\sqrt3\right)\left(3-\sqrt3\right)}{\left(3+\sqrt3\right)\left(3-\sqrt3\right)}=\frac{6-2\sqrt3+3\sqrt3-3}{9-3}=\frac{3+\sqrt3}{6}\)
