\(a,\frac{2xy}{2\sqrt{x}+3\sqrt{y}}=\frac{2xy.\left(2\sqrt{x}-3\sqrt{y}\right)}{\left(2\sqrt{x}+3\sqrt{y}\right)\left(2\sqrt{x}-3\sqrt{y}\right)}=\frac{4x\sqrt{x}y-6xy\sqrt{y}}{2x-3y}\)
\(b,\frac{\sqrt{x}+\sqrt{y}}{2\sqrt{x}}=\frac{\left(\sqrt{x}+\sqrt{y}\right)\sqrt{x}}{2\sqrt{x}\sqrt{x}}=\frac{x+\sqrt{xy}}{2x}\)
\(c,\frac{2}{\sqrt{3}+1}=\frac{2\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}=\frac{2.\left(\sqrt{3}-1\right)}{2}=\sqrt{3}-1\)
\(d,\frac{6}{2\sqrt{3}+\sqrt{2}}=\frac{6\left(2\sqrt{3}-\sqrt{2}\right)}{\left(2\sqrt{3}+\sqrt{2}\right)\left(2\sqrt{3}-\sqrt{2}\right)}=\frac{6\left(2\sqrt{3}-\sqrt{2}\right)}{10}=\frac{6\sqrt{3}-3\sqrt{2}}{5}\)
