\(\frac{2^{19}\times27^3+15\times4^9\times9^4}{6^9\times2^{10}+12^{10}}\)
\(=\frac{2^{19}\times\left(3^3\right)^3+5\times3\times\left(2^2\right)^9\times\left(3^2\right)^4}{\left(2\times3\right)^9\times2^{10}+\left(3\times4\right)^{10}}\)
\(=\frac{2^{19}\times3^9+3\times5\times2^{18}\times3^8}{3^9\times2^9\times2^{10}+3^{10}\times4^{10}}\)
\(=\frac{2^{19}\times3^9+5\times2^{18}\times3^9}{3^9\times2^{19}+3^{10}\times\left(2^2\right)^{10}}\)
\(=\frac{2^{18}\times3^9\times\left(2+5\right)}{3^9\times2^{19}+3^{10}\times2^{20}}\)
\(=\frac{2^{18}\times3^9\times7}{2^{19}\times3^9\times\left(1+3\times2\right)}\)
\(=\frac{7}{2\times\left(1+6\right)}\)
\(=\frac{7}{2\times7}\)
\(=\frac{1}{2}\)
A = 2^19.27^3+15.4^9.9^4 / 6^9.2^10+12^10
= 2^19.3^9 + 5.2^18.3^9 / 3^9.2^19 + 2^20.3^10
= 2^18.3^9 ( 2 + 5 ) / 2^19.3^9.(1 + 2.3)
= (2 + 5) / 2(1 + 6)
= 7 / 2.7
= 1/2
\(\frac{2^{19}.3^9+3^{13}.5.2^{18}}{2^{19}.3^9+2^{20}.3^{10}}=\frac{2^{18}\left(3^9+3^{13}.5\right)}{2^{19}\left(\cdot3^9+2.3^{10}\right)}=\frac{2^{18}.3^9\left(1+3^4.5\right)}{2^{19}.3^9\left(1+2.3\right)}=\frac{1+3^4.5}{2.7}=\frac{406}{14}=29\)
