$\text{a)}{\left(\sqrt{3}+1\right)}^3={\left(\sqrt{3}\right)}^3+3\cdot {\left(\sqrt{3}\right)}^2\cdot 1+3\cdot \sqrt{3}\cdot 1^2+1^3=$  

$=3\sqrt{3}+3\cdot 3\cdot 1+3\cdot \sqrt{3}\cdot 1+1=3\sqrt{3}+9+3\sqrt{3}+1=6\sqrt{3}+10$   

$\text{b)}{\left(\sqrt{2}-1\right)}^3={\left(\sqrt{2}\right)}^3-3\cdot {\left(\sqrt{2}\right)}^2\cdot 1+3\cdot \sqrt{2}\cdot 1^2-1^3=$  

$=2\sqrt{2}-3\cdot 2\cdot 1+3\cdot \sqrt{2}\cdot 1-1=2\sqrt{2}-6+3\sqrt{2}-1=5\sqrt{2}-7$  

$\text{c)}{\left(2\sqrt{2}+\sqrt{3}\right)}^3={\left(2\sqrt{2}\right)}^3+3\cdot {\left(2\sqrt{2}\right)}^2\cdot \sqrt{3}+3\cdot 2\sqrt{2}\cdot {\left(\sqrt{3}\right)}^2+{\left(\sqrt{3}\right)}^3=$  

$=2^3\cdot {\left(\sqrt{2}\right)}^3+3\cdot 2^2\cdot {\left(\sqrt{2}\right)}^2\cdot \sqrt{3}+6\sqrt{2}\cdot 3+3\sqrt{3}=$   

$=8\cdot 2\sqrt{2}+3\cdot 4\cdot 2\cdot \sqrt{3}+18\sqrt{2}+3\sqrt{3}=16\sqrt{2}+24\sqrt{3}+18\sqrt{2}+3\sqrt{3}=$    

$=27\sqrt{3}+34\sqrt{2}$  

$\text{d)}{\left(\sqrt{5}-2\sqrt{3}\right)}^3={\left(\sqrt{5}\right)}^3-3\cdot {\left(\sqrt{5}\right)}^2\cdot 2\sqrt{3}+3\cdot \sqrt{5}\cdot {\left(2\sqrt{3}\right)}^2-{\left(2\sqrt{3}\right)}^3=$  

$=5\sqrt{5}-3\cdot 5\cdot 2\sqrt{3}+3\cdot \sqrt{5}\cdot 2^2\cdot {\left(\sqrt{3}\right)}^2-2^3\cdot {\left(\sqrt{3}\right)}^3=$  

$=5\sqrt{5}-30\sqrt{3}+3\cdot \sqrt{5}\cdot 4\cdot 3-8\cdot 3\sqrt{3}=$  

$=5\sqrt{5}-30\sqrt{3}+36\sqrt{5}-24\sqrt{3}=41\sqrt{5}-54\sqrt{3}$