$$\left. \begin{array} { l } { \frac { 6 } { 3. \sqrt { 2 } - 2 \sqrt { 3 } } = 8 \sqrt { 2 } - a \sqrt { 3 } } \end{array} \right.$$
$a=\frac{5\sqrt{6}}{3}-2\approx 2.082482905$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{\left(3\sqrt{2}-2\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}=8\sqrt{2}-a\sqrt{3}$$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{\left(3\sqrt{2}\right)^{2}-\left(-2\sqrt{3}\right)^{2}}=8\sqrt{2}-a\sqrt{3}$$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{3^{2}\left(\sqrt{2}\right)^{2}-\left(-2\sqrt{3}\right)^{2}}=8\sqrt{2}-a\sqrt{3}$$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{9\left(\sqrt{2}\right)^{2}-\left(-2\sqrt{3}\right)^{2}}=8\sqrt{2}-a\sqrt{3}$$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{9\times 2-\left(-2\sqrt{3}\right)^{2}}=8\sqrt{2}-a\sqrt{3}$$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{18-\left(-2\sqrt{3}\right)^{2}}=8\sqrt{2}-a\sqrt{3}$$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{18-\left(-2\right)^{2}\left(\sqrt{3}\right)^{2}}=8\sqrt{2}-a\sqrt{3}$$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{18-4\left(\sqrt{3}\right)^{2}}=8\sqrt{2}-a\sqrt{3}$$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{18-4\times 3}=8\sqrt{2}-a\sqrt{3}$$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{18-12}=8\sqrt{2}-a\sqrt{3}$$
$$\frac{6\left(3\sqrt{2}+2\sqrt{3}\right)}{6}=8\sqrt{2}-a\sqrt{3}$$
$$3\sqrt{2}+2\sqrt{3}=8\sqrt{2}-a\sqrt{3}$$
$$8\sqrt{2}-a\sqrt{3}=3\sqrt{2}+2\sqrt{3}$$
$$-a\sqrt{3}=3\sqrt{2}+2\sqrt{3}-8\sqrt{2}$$
$$-a\sqrt{3}=-5\sqrt{2}+2\sqrt{3}$$
$$\left(-\sqrt{3}\right)a=2\sqrt{3}-5\sqrt{2}$$
$$\frac{\left(-\sqrt{3}\right)a}{-\sqrt{3}}=\frac{2\sqrt{3}-5\sqrt{2}}{-\sqrt{3}}$$
$$a=\frac{2\sqrt{3}-5\sqrt{2}}{-\sqrt{3}}$$
$$a=\frac{5\sqrt{6}}{3}-2$$
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