$$\frac{2\sqrt{6}-\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}$$
$\frac{4\sqrt{30}}{21}+\frac{3}{7}\approx 1.47185249$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{\left(3\sqrt{5}-2\sqrt{6}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}$$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{\left(3\sqrt{5}\right)^{2}-\left(-2\sqrt{6}\right)^{2}}$$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{3^{2}\left(\sqrt{5}\right)^{2}-\left(-2\sqrt{6}\right)^{2}}$$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{9\left(\sqrt{5}\right)^{2}-\left(-2\sqrt{6}\right)^{2}}$$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{9\times 5-\left(-2\sqrt{6}\right)^{2}}$$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{45-\left(-2\sqrt{6}\right)^{2}}$$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{45-\left(-2\right)^{2}\left(\sqrt{6}\right)^{2}}$$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{45-4\left(\sqrt{6}\right)^{2}}$$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{45-4\times 6}$$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{45-24}$$
$$\frac{\left(2\sqrt{6}-\sqrt{5}\right)\left(3\sqrt{5}+2\sqrt{6}\right)}{21}$$
$$\frac{6\sqrt{6}\sqrt{5}+4\left(\sqrt{6}\right)^{2}-3\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{6}}{21}$$
$$\frac{6\sqrt{30}+4\left(\sqrt{6}\right)^{2}-3\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{6}}{21}$$
$$\frac{6\sqrt{30}+4\times 6-3\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{6}}{21}$$
$$\frac{6\sqrt{30}+24-3\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{6}}{21}$$
$$\frac{6\sqrt{30}+24-3\times 5-2\sqrt{5}\sqrt{6}}{21}$$
$$\frac{6\sqrt{30}+24-15-2\sqrt{5}\sqrt{6}}{21}$$
$$\frac{6\sqrt{30}+9-2\sqrt{5}\sqrt{6}}{21}$$
$$\frac{6\sqrt{30}+9-2\sqrt{30}}{21}$$
$$\frac{4\sqrt{30}+9}{21}$$
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