10.
\(A=\left(\dfrac{3}{4}-81\right)\left(\dfrac{3^2}{5}-81\right)...\left(\dfrac{3^6}{9}-81\right)....\left(\dfrac{3^{2000}}{2003}-81\right)\)
\(=\left(\dfrac{3}{4}-81\right)\left(\dfrac{3^2}{5}-81\right)...\left(\dfrac{729}{9}-81\right)...\left(\dfrac{3^{2000}}{2003}-81\right)\)
mà \(\dfrac{729}{9}-81=0\Rightarrow A=0\)
\(\left(\dfrac{3}{4}-81\right).\left(\dfrac{3^2}{5}-81\right).\left(\dfrac{3^3}{6}-81\right).....\left(\dfrac{3^{2000}}{2003}-81\right)\)
\(=\left(\dfrac{3}{4}-81\right).\left(\dfrac{3^2}{5}-81\right).\left(\dfrac{3^3}{6}-81\right).....\left(\dfrac{3^6}{9}-81\right).....\left(\dfrac{3^{2000}}{2003}-81\right)\)
\(=\left(\dfrac{3}{4}-81\right).\left(\dfrac{3^2}{5}-81\right).\left(\dfrac{3^3}{6}-81\right).....0.....\left(\dfrac{3^{2000}}{2003}-81\right)\)
\(=0\)