$$\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-8\sqrt{3}}=a+b\sqrt{6}$$
$b=-\frac{\sqrt{6}a}{6}-\frac{5\sqrt{6}}{174}-\frac{11}{174}$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{\left(3\sqrt{2}-8\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{\left(3\sqrt{2}\right)^{2}-\left(-8\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{3^{2}\left(\sqrt{2}\right)^{2}-\left(-8\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{9\left(\sqrt{2}\right)^{2}-\left(-8\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{9\times 2-\left(-8\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{18-\left(-8\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{18-\left(-8\right)^{2}\left(\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{18-64\left(\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{18-64\times 3}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{18-192}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+8\sqrt{3}\right)}{-174}=a+b\sqrt{6}$$
$$\frac{3\left(\sqrt{2}\right)^{2}+11\sqrt{2}\sqrt{3}+8\left(\sqrt{3}\right)^{2}}{-174}=a+b\sqrt{6}$$
$$\frac{3\times 2+11\sqrt{2}\sqrt{3}+8\left(\sqrt{3}\right)^{2}}{-174}=a+b\sqrt{6}$$
$$\frac{6+11\sqrt{2}\sqrt{3}+8\left(\sqrt{3}\right)^{2}}{-174}=a+b\sqrt{6}$$
$$\frac{6+11\sqrt{6}+8\left(\sqrt{3}\right)^{2}}{-174}=a+b\sqrt{6}$$
$$\frac{6+11\sqrt{6}+8\times 3}{-174}=a+b\sqrt{6}$$
$$\frac{6+11\sqrt{6}+24}{-174}=a+b\sqrt{6}$$
$$\frac{30+11\sqrt{6}}{-174}=a+b\sqrt{6}$$
$$\frac{-30-11\sqrt{6}}{174}=a+b\sqrt{6}$$
$$-\frac{5}{29}-\frac{11}{174}\sqrt{6}=a+b\sqrt{6}$$
$$a+b\sqrt{6}=-\frac{5}{29}-\frac{11}{174}\sqrt{6}$$
$$b\sqrt{6}=-\frac{5}{29}-\frac{11}{174}\sqrt{6}-a$$
$$\sqrt{6}b=-a-\frac{11\sqrt{6}}{174}-\frac{5}{29}$$
$$\frac{\sqrt{6}b}{\sqrt{6}}=\frac{-a-\frac{11\sqrt{6}}{174}-\frac{5}{29}}{\sqrt{6}}$$
$$b=\frac{-a-\frac{11\sqrt{6}}{174}-\frac{5}{29}}{\sqrt{6}}$$
$$b=-\frac{\sqrt{6}a}{6}-\frac{5\sqrt{6}}{174}-\frac{11}{174}$$
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$a=-\sqrt{6}b-\frac{11\sqrt{6}}{174}-\frac{5}{29}$
