$$\frac{3+\sqrt{5}}{2\sqrt{6}-5\sqrt{3}}$$
$-\frac{2\sqrt{6}}{17}-\frac{2\sqrt{30}}{51}-\frac{5\sqrt{3}}{17}-\frac{5\sqrt{15}}{51}\approx -1.392099381$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{\left(2\sqrt{6}-5\sqrt{3}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}$$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{\left(2\sqrt{6}\right)^{2}-\left(-5\sqrt{3}\right)^{2}}$$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{2^{2}\left(\sqrt{6}\right)^{2}-\left(-5\sqrt{3}\right)^{2}}$$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{4\left(\sqrt{6}\right)^{2}-\left(-5\sqrt{3}\right)^{2}}$$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{4\times 6-\left(-5\sqrt{3}\right)^{2}}$$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{24-\left(-5\sqrt{3}\right)^{2}}$$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{24-\left(-5\right)^{2}\left(\sqrt{3}\right)^{2}}$$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{24-25\left(\sqrt{3}\right)^{2}}$$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{24-25\times 3}$$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{24-75}$$
$$\frac{\left(3+\sqrt{5}\right)\left(2\sqrt{6}+5\sqrt{3}\right)}{-51}$$
$$\frac{6\sqrt{6}+15\sqrt{3}+2\sqrt{5}\sqrt{6}+5\sqrt{5}\sqrt{3}}{-51}$$
$$\frac{6\sqrt{6}+15\sqrt{3}+2\sqrt{30}+5\sqrt{5}\sqrt{3}}{-51}$$
$$\frac{6\sqrt{6}+15\sqrt{3}+2\sqrt{30}+5\sqrt{15}}{-51}$$
$$\frac{-6\sqrt{6}-15\sqrt{3}-2\sqrt{30}-5\sqrt{15}}{51}$$
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