f: \(\sqrt{4+\sqrt{15}}+\sqrt{6-\sqrt{35}}-\sqrt{\dfrac{7}{2}}\)
\(=\dfrac{\sqrt{8+2\sqrt{15}}+\sqrt{12-2\sqrt{35}}-\sqrt{7}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{5}+\sqrt{3}+\sqrt{7}-\sqrt{5}-\sqrt{7}}{\sqrt{2}}=\dfrac{\sqrt{3}}{\sqrt{2}}\)
\(\sqrt{4-\sqrt{15}}-\sqrt{6+\sqrt{35}}+\sqrt{\dfrac{3}{2}}\)
\(=\dfrac{\sqrt{8-2\sqrt{15}}-\sqrt{12+2\sqrt{35}}+\sqrt{3}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{5}-\sqrt{3}-\sqrt{7}-\sqrt{5}+\sqrt{3}}{\sqrt{2}}=-\sqrt{\dfrac{7}{2}}\)
\(B=\left(\sqrt{4+\sqrt{15}}+\sqrt{6-\sqrt{35}}-\sqrt{\dfrac{7}{2}}\right)^2+\left(\sqrt{4-\sqrt{15}}-\sqrt{6+\sqrt{35}}+\sqrt{\dfrac{3}{2}}\right)^2\)
\(=\left(\sqrt{\dfrac{3}{2}}\right)^2+\left(-\sqrt{\dfrac{7}{2}}\right)^2=\dfrac{3}{2}+\dfrac{7}{2}=5\)
c: \(C=\dfrac{3+\sqrt{5}}{\sqrt{2}+\sqrt{3+\sqrt{5}}}+\dfrac{3-\sqrt{5}}{\sqrt{2}-\sqrt{3-\sqrt{5}}}\)
\(=\dfrac{\sqrt{2}\left(3+\sqrt{5}\right)}{2+\sqrt{6+2\sqrt{5}}}+\dfrac{\sqrt{2}\left(3-\sqrt{5}\right)}{2-\sqrt{6-2\sqrt{5}}}\)
\(=\dfrac{\sqrt{2}\left(3+\sqrt{5}\right)}{2+\sqrt{5}+1}+\dfrac{\sqrt{2}\left(3-\sqrt{5}\right)}{2-\left(\sqrt{5}-1\right)}\)
\(=\dfrac{\sqrt{2}\left(3+\sqrt{5}\right)}{3+\sqrt{5}}+\dfrac{\sqrt{2}\left(3-\sqrt{5}\right)}{3-\sqrt{5}}=\sqrt{2}+\sqrt{2}=2\sqrt{2}\)
e: \(E=\dfrac{4\cdot\sqrt{3-2\sqrt{2}}+10}{\left(1+\sqrt{2}\right)\left(3+\sqrt{2}\right)+1}\)
\(=\dfrac{4\cdot\sqrt{\left(\sqrt{2}-1\right)^2}+10}{3+\sqrt{2}+3\sqrt{2}+2+1}\)
\(=\dfrac{4\left(\sqrt{2}-1\right)+10}{4\sqrt{2}+6}=\dfrac{4\sqrt{2}+6}{4\sqrt{2}+6}=1\)
f: \(F=\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)
\(=\sqrt{13+30\sqrt{2+\sqrt{\left(2\sqrt{2}+1\right)^2}}}\)
\(=\sqrt{13+30\sqrt{2+\left(2\sqrt{2}+1\right)}}\)
\(=\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\)
\(=\sqrt{13+30\left(\sqrt{2}+1\right)}=\sqrt{43+30\sqrt{2}}\)
\(=\sqrt{25+2\cdot5\cdot3\sqrt{2}+18}=\sqrt{\left(5+3\sqrt{2}\right)^2}=5+3\sqrt{2}\)
