$$\frac{2\sqrt{2}-\sqrt{5}}{\sqrt{5}+\sqrt{2}}$$
$\sqrt{10}-3\approx 0.16227766$
$$\frac{\left(2\sqrt{2}-\sqrt{5}\right)\left(\sqrt{5}-\sqrt{2}\right)}{\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right)}$$
$$\frac{\left(2\sqrt{2}-\sqrt{5}\right)\left(\sqrt{5}-\sqrt{2}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$
$$\frac{\left(2\sqrt{2}-\sqrt{5}\right)\left(\sqrt{5}-\sqrt{2}\right)}{5-2}$$
$$\frac{\left(2\sqrt{2}-\sqrt{5}\right)\left(\sqrt{5}-\sqrt{2}\right)}{3}$$
$$\frac{2\sqrt{2}\sqrt{5}-2\left(\sqrt{2}\right)^{2}-\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}}{3}$$
$$\frac{2\sqrt{10}-2\left(\sqrt{2}\right)^{2}-\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}}{3}$$
$$\frac{2\sqrt{10}-2\times 2-\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}}{3}$$
$$\frac{2\sqrt{10}-4-\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}}{3}$$
$$\frac{2\sqrt{10}-4-5+\sqrt{5}\sqrt{2}}{3}$$
$$\frac{2\sqrt{10}-9+\sqrt{5}\sqrt{2}}{3}$$
$$\frac{2\sqrt{10}-9+\sqrt{10}}{3}$$
$$\frac{3\sqrt{10}-9}{3}$$
$$\sqrt{10}-3$$
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