$$\frac{5+\sqrt{3}}{3+\sqrt{6}}$$
$-\frac{5\sqrt{6}}{3}+\sqrt{3}+5-\sqrt{2}\approx 1.235354341$
$$\frac{\left(5+\sqrt{3}\right)\left(3-\sqrt{6}\right)}{\left(3+\sqrt{6}\right)\left(3-\sqrt{6}\right)}$$
$$\frac{\left(5+\sqrt{3}\right)\left(3-\sqrt{6}\right)}{3^{2}-\left(\sqrt{6}\right)^{2}}$$
$$\frac{\left(5+\sqrt{3}\right)\left(3-\sqrt{6}\right)}{9-6}$$
$$\frac{\left(5+\sqrt{3}\right)\left(3-\sqrt{6}\right)}{3}$$
$$\frac{15-5\sqrt{6}+3\sqrt{3}-\sqrt{3}\sqrt{6}}{3}$$
$$\frac{15-5\sqrt{6}+3\sqrt{3}-\sqrt{3}\sqrt{3}\sqrt{2}}{3}$$
$$\frac{15-5\sqrt{6}+3\sqrt{3}-3\sqrt{2}}{3}$$
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