$$\frac{ 13 }{ 2 \sqrt{ 3 } + \sqrt{ 11 } }$$
$26\sqrt{3}-13\sqrt{11}\approx 1.917198722$
$$\frac{13\left(2\sqrt{3}-\sqrt{11}\right)}{\left(2\sqrt{3}+\sqrt{11}\right)\left(2\sqrt{3}-\sqrt{11}\right)}$$
$$\frac{13\left(2\sqrt{3}-\sqrt{11}\right)}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{11}\right)^{2}}$$
$$\frac{13\left(2\sqrt{3}-\sqrt{11}\right)}{2^{2}\left(\sqrt{3}\right)^{2}-\left(\sqrt{11}\right)^{2}}$$
$$\frac{13\left(2\sqrt{3}-\sqrt{11}\right)}{4\left(\sqrt{3}\right)^{2}-\left(\sqrt{11}\right)^{2}}$$
$$\frac{13\left(2\sqrt{3}-\sqrt{11}\right)}{4\times 3-\left(\sqrt{11}\right)^{2}}$$
$$\frac{13\left(2\sqrt{3}-\sqrt{11}\right)}{12-\left(\sqrt{11}\right)^{2}}$$
$$\frac{13\left(2\sqrt{3}-\sqrt{11}\right)}{12-11}$$
$$\frac{13\left(2\sqrt{3}-\sqrt{11}\right)}{1}$$
$$13\left(2\sqrt{3}-\sqrt{11}\right)$$
$$26\sqrt{3}-13\sqrt{11}$$
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