$$\frac{5+2\sqrt{3}}{7+\sqrt{2}}$$
$\frac{14\sqrt{3}+35-2\sqrt{6}-5\sqrt{2}}{47}\approx 1.005929021$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-\sqrt{2}\right)}{\left(7+\sqrt{2}\right)\left(7-\sqrt{2}\right)}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-\sqrt{2}\right)}{7^{2}-\left(\sqrt{2}\right)^{2}}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-\sqrt{2}\right)}{49-2}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-\sqrt{2}\right)}{47}$$
$$\frac{35-5\sqrt{2}+14\sqrt{3}-2\sqrt{3}\sqrt{2}}{47}$$
$$\frac{35-5\sqrt{2}+14\sqrt{3}-2\sqrt{6}}{47}$$
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