a, \(\frac{2^{30}.5^7+2^{13}.5^{27}}{2^{27}.5^7+2^{10}.5^{27}}=\frac{2^{13}.\left(2^{17}.5^7+5^{27}\right)}{2^{10}.\left(2^{17}.5^7+5^{27}\right)}=\frac{2^{13}}{2^{10}}=2^3=8\).
b, \(\frac{81.2^2+3^4+20.9^2}{16.3^2+45+2^2.9}=\frac{3^4.2^2+3^4+20.3^4}{16.3^2+3^2.5+2^2.3^2}=\frac{3^4.\left(2^2+1+20\right)}{3^2.\left(16+5+2^2\right)}=\frac{3^4.25}{3^2.25}=\frac{3^4}{3^2}=3^2=9\)