a: \(\sqrt{6-3\sqrt{3}}+\sqrt{2-\sqrt{3}}\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{12-6\sqrt{3}}+\sqrt{4-2\sqrt{3}}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{\left(3-\sqrt{3}\right)^2}+\sqrt{\left(\sqrt{3}-1\right)^2}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(3-\sqrt{3}+\sqrt{3}-1\right)=\dfrac{2}{\sqrt{2}}=\sqrt{2}\)
b: \(\sqrt{15+5\sqrt{5}}-\sqrt{3-\sqrt{5}}\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{30+10\sqrt{5}}-\sqrt{6-2\sqrt{5}}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{\left(5+\sqrt{5}\right)^2}-\sqrt{\left(\sqrt{5}-1\right)^2}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(5+\sqrt{5}-\sqrt{5}+1\right)=\dfrac{1}{\sqrt{2}}\cdot6=3\sqrt{2}\)
c: \(\sqrt{24-3\sqrt{15}}-\sqrt{36-9\sqrt{15}}\)
\(=\dfrac{1}{\sqrt[]{2}}\left(\sqrt{48-6\sqrt{15}}-\sqrt{72-18\sqrt{15}}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{\left(3\sqrt{5}-\sqrt{3}\right)^2}-\sqrt{\left(3\sqrt[]{5}-3\sqrt{3}\right)^2}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(3\sqrt{5}-\sqrt{3}-3\sqrt{5}+3\sqrt{3}\right)=\dfrac{2\sqrt{3}}{\sqrt{2}}=\sqrt{6}\)
d: \(\sqrt{2-\sqrt{3}}-\sqrt{2+\sqrt{3}}\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{4-2\sqrt{3}}-\sqrt{4+2\sqrt{3}}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{3}-1\right)^2}-\sqrt{\left(\sqrt{3}+1\right)^2}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{3}-1-\sqrt{3}-1\right)=-\dfrac{2}{\sqrt{2}}=-\sqrt{2}\)
e: \(\sqrt{3-\sqrt{5}}-\sqrt{3+\sqrt{5}}\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{6-2\sqrt{5}}-\sqrt{6+2\sqrt{5}}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{5}-1\right)^2}-\sqrt{\left(\sqrt{5}+1\right)^2}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{5}-1-\sqrt{5}-1\right)=-\dfrac{2}{\sqrt{2}}=-\sqrt{2}\)
f: \(\sqrt{9-\sqrt{17}}+\sqrt{9+\sqrt{17}}\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{18-2\sqrt{17}}+\sqrt{18+2\sqrt{17}}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{17}-1\right)^2}+\sqrt{\left(\sqrt{17}+1\right)^2}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{17}-1+\sqrt{17}+1\right)=\dfrac{2\sqrt{17}}{\sqrt{2}}=\sqrt{34}\)
g: \(\sqrt{7+\sqrt{13}}-\sqrt{7-\sqrt{13}}\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{14+2\sqrt{13}}-\sqrt{14-2\sqrt{13}}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{13}+1\right)^2}-\sqrt{\left(\sqrt{13}-1\right)^2}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{13}+1-\sqrt{13}+1\right)=\dfrac{1}{\sqrt{2}}\cdot2=\sqrt{2}\)
h: \(\sqrt{12-3\sqrt{7}}-\sqrt{12+3\sqrt{7}}\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{24-6\sqrt{7}}-\sqrt{24+6\sqrt{7}}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{21-2\cdot\sqrt{21}\cdot\sqrt{3}+3}-\sqrt{21+2\cdot\sqrt{21}\cdot\sqrt{3}+3}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{21}-\sqrt{3}\right)^2}-\sqrt{\left(\sqrt{21}+\sqrt{3}\right)^2}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt{21}-\sqrt{3}-\sqrt{21}-\sqrt{3}\right)=-\dfrac{2\sqrt{3}}{\sqrt{2}}=-\sqrt{6}\)
