\(\dfrac{1}{\sqrt{10}+\sqrt{15}+\sqrt{14}+\sqrt{21}}\)
\(=\dfrac{1}{\sqrt{5}\left(\sqrt{2}+\sqrt{3}\right)+\sqrt{7}\left(\sqrt{2}+\sqrt{3}\right)}\)
\(=\dfrac{1}{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{5}+\sqrt{7}\right)}\)
\(=\dfrac{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\left(3-2\right)\left(7-5\right)}=\dfrac{\sqrt{21}-\sqrt{15}-\sqrt{14}+\sqrt{10}}{2}\)
\(\dfrac{3+2\sqrt{3}}{\sqrt{3}}+\dfrac{2+\sqrt{2}}{\sqrt{2}+1}-\left(\sqrt{2}+\sqrt{3}\right)\)
\(=\dfrac{\sqrt{3}\left(2+\sqrt{3}\right)}{\sqrt{3}}+\dfrac{\sqrt{2}\left(\sqrt{2}+1\right)}{\sqrt{2}+1}-\left(\sqrt{2}+\sqrt{3}\right)\)
\(=\sqrt{3}+\sqrt{2}-\sqrt{3}-\sqrt{2}\)
=0
