\(\sqrt[3]{10+6\sqrt{3}}.\left(\sqrt{3}-1\right)\)
\(=\sqrt[3]{1+3\sqrt{3}+9+3\sqrt{3}}\left(\sqrt{3}-1\right)\)
\(=\sqrt[3]{\left(1+\sqrt{3}\right)^3}\left(\sqrt{3}-1\right)\)
\(=\left(1+\sqrt{3}\right)\left(\sqrt{3}-1\right)\)
\(=\sqrt{3}^2-1^2=3-1=2\)