\(\dfrac{3}{2\sqrt{2}-3\sqrt{3}}-\dfrac{3}{2\sqrt{2}+3\sqrt{3}}\)
\(=\dfrac{3\left(2\sqrt{2}+3\sqrt{3}\right)}{\left(2\sqrt{2}-3\sqrt{3}\right)\left(2\sqrt{2}+3\sqrt{3}\right)}-\dfrac{3\left(2\sqrt{2}-3\sqrt{3}\right)}{\left(2\sqrt{2}-3\sqrt{3}\right)\left(2\sqrt{2}+3\sqrt{3}\right)}\)
\(=\dfrac{3\left(2\sqrt{2}+3\sqrt{3}-2\sqrt{2}+3\sqrt{3}\right)}{\left(2\sqrt{2}-3\sqrt{3}\right)\left(2\sqrt{2}+3\sqrt{3}\right)}\)
\(=\dfrac{3\cdot6\sqrt{3}}{\left(2\sqrt{2}\right)^2-\left(3\sqrt{3}\right)^2}\)
\(=\dfrac{18\sqrt{3}}{8-27}\)
\(=-\dfrac{18\sqrt{3}}{19}\)
#Toru
\(\dfrac{3}{2\sqrt{2}-3\sqrt{3}}-\dfrac{3}{2\sqrt{2}+3\sqrt{3}}\\ =\dfrac{3\left(2\sqrt{2}+3\sqrt{3}\right)}{\left(2\sqrt{2}-3\sqrt{3}\right)\left(2\sqrt{2}+3\sqrt{3}\right)}-\dfrac{3\left(2\sqrt{2}-3\sqrt{3}\right)}{\left(2\sqrt{2}+3\sqrt{3}\right)\left(2\sqrt{2}-3\sqrt{3}\right)}\\ =\dfrac{6\sqrt{2}+9\sqrt{3}-6\sqrt{2}+9\sqrt{3}}{8-6\sqrt{6}+6\sqrt{6}-27}\\ =\dfrac{18\sqrt{3}}{-19}\\ =\dfrac{-18\sqrt{3}}{19}\)
