[tex]\dfrac{(a^{7}b^{-2}):b^{3}}{(a^{8}:a^{-3})\cdot(b^{-2})^{2}}=\dfrac{a^{7}b^{-5}}{a^{11}\cdot b^{-4}}}=a^{-4}b^{-1}\\\\\\dla \ \ a = \frac{3}{4}, \ \ b = \frac{8}{9}\\\\\\a^{-4}b^{-1} = (\dfrac{3}{4})^{-4}\cdot(\dfrac{8}{9})^{-1}=(\dfrac{4}{3})^{4}\cdot\dfrac{9}{8}=(\dfrac{2^{2}}{3})^{4}\cdot\dfrac{3^{2}}{2^{3}} = \dfrac{2^{8}}{3^{4}}\cdot\dfrac{3^{2}}{2^{3}}=2^{5}\cdot3^{-2} = 32\cdot(\dfrac{1}{3})^{2}=\\\\\\=32\cdot\dfrac{1}{9}} = \dfrac{32}{9} = 3\dfrac{5}{9}[/tex]
Wyjaśnienie:
[tex]a^{m}\cdot a^{n} = a^{m+n}\\\\a^{m}:a^{n} = a^{m-n}\\\\(a^{m})^{n} = a^{m\cdot n}\\\\(\frac{a}{b})^{-n} = (\frac{b}{a})^{n}, \ \ dla \ a \neq 0[/tex]
