$a)\frac{6^8\cdot {\left(\frac{1}{3}\right)}^8}{{20}^6:{10}^6}=\frac{{\left(6\cdot \frac{1}{3}\right)}^8}{{\left(20:10\right)}^6}=\frac{2^8}{2^6}=2^{8-6}=2^2=4$

 

$b)\frac{{45}^6:{\left(5^2\right)}^3}{3^1\cdot 3^2\cdot 3^3}=\frac{{45}^6:5^6}{3^{1+2+3}}=\frac{{\left(45:5\right)}^6}{3^6}=\frac{9^6}{3^3}={\left(\frac{9}{3}\right)}^6=3^6=9^3=9\cdot 9\cdot 9=81\cdot 9=729$

 

$c){\left[{\left(\frac{1}{2}\right)}^{10}\cdot 6^{10}\right]}^2:{\left[{12}^6\cdot {\left(\frac{1}{4}\right)}^6\right]}^3={\left[{\left(\frac{1}{2}\cdot 6\right)}^{10}\right]}^2:{\left[{\left(12\cdot \frac{1}{4}\right)}^6\right]}^3=$

$={\left[3^{10}\right]}^2:{\left[3^6\right]}^3=3^{10\cdot 2}:3^{6\cdot 3}=3^{20}:3^{18}=3^{20-18}=3^2=9$