4^5*5^6=4^5*5*5^5=(4*5)^5*5=20^5*5=2^5*10^5*5=160*10^5=16000000
(\frac{3}{5})^5*(\frac{5}{3})^6=\frac{3^5}{5^5}*\frac{5^6}{3^6}=3^{5-6}*5^{6-5}=3^{-1}*5=\frac{1}{3}*5=\frac{5}{3}=1\frac{2}{3}
(\frac{4}{9})^7:(\frac{2}{9})^8=\frac{4^7}{9^7}:\frac{(2^2)^4}{9^8}=\frac{4^7}{9^7}*\frac{9^8}{4^4}=4^{7-4}*9^{8-7}=4^3*9=64*9=576
(1\frac{4}{9})^5:(4\frac{1}{3})^4=(\frac{13}{9})^5:(\frac{13}{3})^4=\frac{13^5}{9^5}*\frac{9^2}{13^4}=13^{5-4}*9^{2-5}=13*9^{-3}=13*\frac{1}{9^3}=\frac{13}{729}