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\(\frac{4^6.9^5+6^9.120}{8^4.3^{12}-6^{11}}=\frac{\left(2^2\right)^6.\left(3^2\right)^5+\left(2.3\right)^9.2^3.3.5}{\left(2^3\right)^4.3^{12}-\left(2.3\right)^{11}}\)
\(=\frac{2^{12}.3^{10}+2^{12}.3^{10}.5}{2^{12}.3^{12}-2^{11}.3^{11}}\)
\(=\frac{2^{12}.3^{10}}{2^{11}.3^{11}}.\frac{1+5}{2.3-1}\)
\(=\frac{2^{12}.3^{10}}{2^{11}.3^{11}}.\frac{6}{5}\)
\(=\frac{2}{3}.\frac{6}{5}\)
\(=\frac{4}{5}\)
\(A=\frac{4^6\cdot9^5+6^9\cdot120}{8^4\cdot3^{12}-6^{11}}=\frac{\left(2^2\right)^6\cdot\left(3^2\right)^5+\left(2\cdot3\right)^9\cdot2^3\cdot3\cdot5}{\left(2^3\right)^4\cdot3^{12}-\left(2\cdot3\right)^{11}}=\frac{2^{12}\cdot3^{10}+2^9\cdot3^9\cdot2^3\cdot3\cdot5}{2^{12}\cdot3^{12}-2^{11}\cdot3^{11}}=\frac{2^{12}\cdot3^{10}+2^{12}\cdot3^{10}\cdot5}{2^{12}\cdot3^{12}-2^{11}\cdot3^{11}}=\frac{2^{12}\cdot3^{10}\left(1+5\right)}{2^{11}\cdot3^{11}\left(2\cdot3-1\right)}=\frac{2\left(1+5\right)}{3\left(6-1\right)}=\frac{2\cdot6}{3\cdot5}=\frac{2\cdot2}{5}=\frac{4}{5}\)
\(\approx GOOD\)\(LUCK\approx\)
\(=\frac{2^{12}.3^{10}+2^9.3^9.2^3.3.5}{-2^{12}.3^{12}+2^{11}.3^{11}}=\frac{2^{12}.3^{10}+2^{12}.3^{10}.5}{2^{11}.3^{11}\left(-2.3+1\right)}=\frac{2^{12}.3^{10}\left(1+5\right)}{2^{11}.3^{11}.\left(-5\right)}=\frac{2.6}{3.\left(-5\right)}=-\frac{4}{5}\)
\(\frac{4^6.9^5+6^9.120}{8^4.3^2-6^{11}}\)
\(=\frac{2^{12}.3^{10}+2^9.3^9.2^3.3.5}{2^{12}.3^2-2^{11}.3^{11}}\)
\(=\frac{2^{12}.3^{10}+2^{12}.3^{10}.5}{2^{11}.3^2.\left(2-3^9\right)}\)
\(=\frac{2^{12}.3^{10}.\left(1+5\right)}{2^{11}.3^2.\left(2-3^9\right)}\)
\(=\frac{2.3^8.6}{2-3^9}\)
\(=\frac{2.2.3.3^8}{2-3^9}\)
\(=\frac{2^2.3^9}{2-3^9}\)