$a)\left|AB\right|=\sqrt{{\left(5-2\right)}^2+{\left(0-\left(-4\right)\right)}^2}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$  

 

$b)\left|AB\right|=\sqrt{{\left(1-\left(-1\right)\right)}^2+{\left(5-4\right)}^2}=\sqrt{2^2+1^2}=\sqrt{4+1}=\sqrt{5}$  

 

$c)\left|AB\right|=\sqrt{{\left(-2-1\right)}^2+{\left(3+\sqrt{2}-\sqrt{2}\right)}^2}=\sqrt{{\left(-3\right)}^2+3^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$  

 

$d)\left|AB\right|=\sqrt{{\left(\sqrt{2}-3\right)}^2+{\left(-\sqrt{3}-\sqrt{6}\right)}^2}=\sqrt{2-6\sqrt{2}+9+3+2\sqrt{18}+6}=\sqrt{20-6\sqrt{2}+6\sqrt{2}}=\sqrt{20}=2\sqrt{5}$