\(\frac{\sqrt{\sqrt{4+\sqrt{15}}+\sqrt{5-\sqrt{21}}}}{\sqrt{6+\sqrt{35}}}\)+\(\sqrt{\frac{1}{4-2\sqrt{3}}}\)-\(\sqrt{\frac{1}{4+2\sqrt{3}}}\)
=\(\frac{\sqrt{\sqrt{\frac{1}{2}\left(8+2\sqrt{15}\right)}+\sqrt{\frac{1}{2}\left(10-2\sqrt{21}\right)}}}{\sqrt{\frac{1}{2}\left(12+2\sqrt{35}\right)}}\)+\(\sqrt{\frac{1}{3-2\sqrt{3}.1+1}}\)-\(\sqrt{\frac{1}{3+2\sqrt{3}.1+1}}\)
=\(\frac{\sqrt{\sqrt{\frac{1}{2}\left(5+2\sqrt{5}.\sqrt{3}+3\right)}+\sqrt{\frac{1}{2}\left(7-2\sqrt{7}.\sqrt{3}+3\right)}}}{\sqrt{\frac{1}{2}\left(7+2\sqrt{7}.\sqrt{5}+5\right)}}\)+\(\sqrt{\frac{1}{\left(\sqrt{3}-1\right)^2}}\)-\(\sqrt{\frac{1}{\left(\sqrt{3}+1\right)^2}}\)
=\(\frac{\sqrt{\sqrt{\frac{1}{2}\left(\sqrt{5}+\sqrt{3}\right)^2}+\sqrt{\frac{1}{2}\left(\sqrt{7}-\sqrt{3}\right)^2}}}{\sqrt{\frac{1}{2}\left(\sqrt{7}+\sqrt{5}\right)^2}}\)+\(\frac{1}{\sqrt{3}-1}\)-\(\frac{1}{\sqrt{3}+1}\)
=\(\frac{\sqrt{\sqrt{\frac{1}{2}}.\left(\sqrt{5}+\sqrt{3}\right)+\sqrt{\frac{1}{2}}.\left(\sqrt{7}-\sqrt{3}\right)}}{\sqrt{\frac{1}{2}}.\left(\sqrt{7}+\sqrt{5}\right)}\)+\(\frac{\sqrt{3}+1-\sqrt{3}+1}{3-1}\)
=\(\frac{\sqrt{\sqrt{\frac{1}{2}}.\left(\sqrt{7}+\sqrt{5}\right)}}{\sqrt{\frac{1}{2}}.\left(\sqrt{7}+\sqrt{5}\right)}\)+1
=\(\frac{1}{\sqrt{\sqrt{\frac{1}{2}}.\left(\sqrt{7}+\sqrt{5}\right)}}\)+1
