$a)\frac{1}{3}\cdot \frac{1}{5}=\frac{1}{15}$

$b)\frac{2}{{\sout{7}}^1}\cdot \frac{{\sout{14}}^2}{25}=\frac{4}{25}$

$c)1\frac{1}{2}\cdot 4=\frac{3}{{\sout{2}}^1}\cdot {\sout{4}}^2=6$

$d)12\cdot 4\frac{5}{8}={\sout{12}}^3\cdot \frac{37}{{\sout{8}}^2}=\frac{111}{2}=55\frac{1}{2}$

$e)2\frac{5}{6}\cdot \frac{3}{17}=\frac{{\sout{17}}^1}{6}\cdot \frac{3}{{\sout{17}}^1}=\frac{3}{6}=\frac{1}{2}$

$f)11\frac{2}{3}\cdot \frac{3}{7}=\frac{35}{{\sout{3}}^1}\cdot \frac{{\sout{3}}^1}{7}=\frac{35}{7}=\frac{5}{1}=5$