$$\frac{4}{2\sqrt{5}+3\sqrt{2}}$$
$2\sqrt{2}\left(\sqrt{10}-3\right)\approx 0.458990536$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{\left(2\sqrt{5}+3\sqrt{2}\right)\left(2\sqrt{5}-3\sqrt{2}\right)}$$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{\left(2\sqrt{5}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{2^{2}\left(\sqrt{5}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{4\left(\sqrt{5}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{4\times 5-\left(3\sqrt{2}\right)^{2}}$$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{20-\left(3\sqrt{2}\right)^{2}}$$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{20-3^{2}\left(\sqrt{2}\right)^{2}}$$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{20-9\left(\sqrt{2}\right)^{2}}$$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{20-9\times 2}$$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{20-18}$$
$$\frac{4\left(2\sqrt{5}-3\sqrt{2}\right)}{2}$$
$$2\left(2\sqrt{5}-3\sqrt{2}\right)$$
$$4\sqrt{5}-6\sqrt{2}$$
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