$$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$$
$11-6\sqrt{3}\approx 0.607695155$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{7^{2}-\left(4\sqrt{3}\right)^{2}}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-\left(4\sqrt{3}\right)^{2}}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-4^{2}\left(\sqrt{3}\right)^{2}}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\left(\sqrt{3}\right)^{2}}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\times 3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-48}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{1}$$
$$\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)$$
$$35-20\sqrt{3}+14\sqrt{3}-8\left(\sqrt{3}\right)^{2}$$
$$35-6\sqrt{3}-8\left(\sqrt{3}\right)^{2}$$
$$35-6\sqrt{3}-8\times 3$$
$$35-6\sqrt{3}-24$$
$$11-6\sqrt{3}$$
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