$a)\frac{1}{{\sout{6}}^3}\cdot {\sout{20}}^{10}=\frac{10}{3}=3\frac{1}{3}$

$b)\frac{3}{{\sout{5}}^1}\cdot {\sout{30}}^6=18$

$c)2\frac{3}{4}\cdot 20=\frac{11}{{\sout{4}}^1}\cdot {\sout{20}}^5=55$

$d)\frac{1}{7}\cdot 10=\frac{10}{7}=1\frac{3}{7}$

$e)\frac{3}{5}\cdot 18=\frac{3\cdot 18}{5}=\frac{54}{5}=10\frac{4}{5}$

$f)2\frac{1}{3}\cdot 20=\frac{7}{3}\cdot 20=\frac{140}{3}=46\frac{2}{3}$