a)
\frac{5x^3*x^{-6}:x^{-9}}{x^{-4}:x}=\frac{5x^{-3}:x^{-9}}{x^{-5}}=\frac{5x^{-3-(-9)}}{x^{-5}}=\frac{5x^6}{x^{-5}}=5x^{6-(-5)}=5x^{11}
x\ne 0
b)
\frac{(-r^2)^3:(-r^4)^2}{r^{-2}*(r^{-1})^{-4}}=\frac{-r^6:(-r^8)}{r^{-2}*r^4}=\frac{-r^{-2}}{r^2}=-r^{-2-2}=-r^{-4}=-\frac{1}{r^4}
r\ne 0
c)
\frac{(a^2b)^{-1}:(a^3b)}{a^3b^{-2}}=\frac{a^{-2}b^{-1}}{a^3b^{-2}}*\frac{1}{a^3b}=\frac{a^{-2}b^{-1}}{a^6b^{-1}}=a^{-2-6}=a^{-8}=\frac{1}{a^8}
a\ne 0 , b\ne 0
d)
\frac{u^2v^3*(u^{-1}v^{-2})^2}{(u^3v)^{-1}:u^{-2}}=\frac{u^2v^3*u^{-2}v^{-4}}{u^{-3}v^{-1}:u^{-2}}=\frac{u^{2-2}v^{3-4}}{u^{-3-(-2)}*v^{-1}}=\frac{u^0v^{-1}}{u^{-1}v^{-1}}=\frac{1}{u^{-1}}=u
u\ne 0 , v \ne 0
e)
(\frac{k}{l^2})^{-1}*(\frac{l^3}{m})^{-2}:(km)^{-3}=\frac{k^{-1}}{l^{-2}}*\frac{l^{-6}}{m^{-2}}:\frac{1}{k^3m^3}=\frac{k^{-1}l^{-6}*k^3m^3}{l^{-2}m^{-2}}=
k^{-1+3}*l^{-6-(-2)}*m^{3-(-2)}=k^2*l^{-4}*m^5=\frac{k^2m^5}{l^4}
k\ne 0 , l \ne 0 , m\ne 0
f)
\frac{ab^2c*(a^3b^2c)^{-1}}{(a^{-1}bc^2)^2}=\frac{ab^2c*a^{-3}b^{-2}c^{-1}}{a^{-2}b^2c^4}=\frac{a^{-2}b^0c^0}{a^{-2}b^2c^4}=a^{-2-(-2)}*b^{0-2}*c^{0-4}=a^0b^{-2}c^{-4}=1*\frac{1}{b^2}*\frac{1}{c^4}=\frac{1}{b^2c^4}
a\ne0 , b\ne 0 , c\ne 0