a: \(2024-\left\{2023-\left[11\cdot3^2-\left(5\cdot2^4-42\right)\right]\right\}\)
\(=2024-2023+\left[11\cdot9-\left(5\cdot16-42\right)\right]\)
\(=1+99-\left(80-42\right)\)
=100-38
=62
b: \(\dfrac{2^{12}\cdot3^5-4^6\cdot9^2}{\left(2^2\cdot3\right)^6+8^4\cdot3^5}+\dfrac{5^{10}\cdot49^2-25^5\cdot7^3}{\left(125\cdot7\right)^3+5^9\cdot14^3}\)
\(=\dfrac{2^{12}\cdot3^5-2^{12}\cdot3^4}{2^{12}\cdot3^6+2^{12}\cdot3^5}+\dfrac{5^{10}\cdot7^4-5^{10}\cdot7^3}{5^9\cdot7^3+5^9\cdot7^3\cdot2^3}\)
\(=\dfrac{2^{12}\cdot3^4\left(3-1\right)}{2^{12}\cdot3^5\left(3+1\right)}+\dfrac{5^{10}\cdot7^3\left(7-1\right)}{5^9\cdot7^3\left(1+2^3\right)}\)
\(=\dfrac{-2}{3\cdot4}+\dfrac{6\cdot5}{9}\)
\(=\dfrac{-1}{6}+\dfrac{10}{3}=\dfrac{-1}{6}+\dfrac{20}{6}=\dfrac{19}{6}\)
c: \(A=1+3^2+3^4+3^6+...+3^{2022}\)
=>\(9\cdot A=3^2+3^4+3^6+...+3^{2024}\)
=>\(9A-A=3^2+3^4+3^6+...+3^{2024}-1-3^2-...-3^{2022}\)
=>\(8A=3^{2024}-1\)
=>\(A=\dfrac{3^{2024}-1}{8}\)
