\(A=\left(\dfrac{3}{4}-81\right)\left(\dfrac{3^2}{5}-81\right)...\left(\dfrac{3^{2000}}{2003}-81\right)\\ A=\left(\dfrac{3}{4}-81\right)\left(\dfrac{3^2}{5}-81\right)...\left(\dfrac{3^6}{9}-81\right)...\left(\dfrac{3^{2000}}{2003}-81\right)\\ A=\left(\dfrac{3}{4}-81\right)\left(\dfrac{3^2}{5}-81\right)...\cdot0\cdot...\left(\dfrac{3^{2000}}{2003}-81\right)\\ A=0\)
\(\left(\dfrac{3}{4}-81\right)\left(\dfrac{3^2}{5}-81\right)\left(\dfrac{3^3}{6}-81\right)...\left(\dfrac{3^{2000}}{2003}-81\right)\)
\(=\left(\dfrac{3}{4}-81\right)\left(\dfrac{3^2}{5}-81\right)...\left(\dfrac{3^6}{9}-81\right)...\left(\dfrac{3^{2000}}{2003}-81\right)\)
\(=\left(\dfrac{3}{4}-81\right)\left(\dfrac{3^2}{5}-81\right)...\left(81-81\right)...\left(\dfrac{3^{2000}}{2003}-81\right)\)
\(=\left(\dfrac{3}{4}-81\right)\left(\dfrac{3^2}{5}-81\right)...0...\left(\dfrac{3^{2000}}{2003}-81\right)=0\)
