\({1 \over x^{4}} = x^{-4}\)

\({4 a^{5} \over 7 a^{9}} = {4 \over 7} ⋅ {a^{5} \over a^{9}} = {4 \over 7} ⋅ a^{5 - 9} = {4 \over 7} a^{-4}\)

\({p^{4} \over p^{-5}} = p^{4 - -5} = p^{9}\)

\(x^{4} ⋅ x^{-7} = x^{4 + -7} = x^{-3}\)

\((a^{8})^{-7} = a^{8 ⋅ -7} = a^{-56}\)

\(x^{6} ⋅ {1 \over x^{7}} = x^{6} ⋅ x^{-7} = x^{6 + -7} = x^{-1}\)

\({({1 \over x^{4}}) \over x^{3}} = {x^{-4} \over x^{3}} = x^{-4 - 3} = x^{-7}\)

\({a^{3} \over ({1 \over a^{4}})} = {a^{3} \over a^{-4}} = a^{3 - -4} = a^{7}\)

\({8 x^{4} y^{4} \over 3 x y^{9}} = {8 \over 3} ⋅ {x^{4} \over x^{1}} ⋅ {y^{4} \over y^{9}} = {8 \over 3} ⋅ x^{4 - 1} ⋅ x^{4 - 9} = 2\frac{2}{3} x^{3} y^{-5}\)

\({a^{9} \over a^{0}} = a^{9 - 0} = a^{9}\)

\(a^{9} ⋅ \sqrt[3]{a} = a^{9} ⋅ a^{\frac{1}{3}} = a^{9 + \frac{1}{3}} = a^{9\frac{1}{3}}\)

\(x^{6} ⋅ \sqrt[9]{x^{7}} = x^{6} ⋅ x^{\frac{7}{9}} = x^{6 + \frac{7}{9}} = x^{6\frac{7}{9}}\)

\({a^{3} \over \sqrt[5]{a^{3}}} = {a^{3} \over a^{\frac{3}{5}}} = a^{3 - \frac{3}{5}} = a^{2\frac{2}{5}}\)

\({1 \over x^{6}} ⋅ \sqrt[9]{x^{2}} = x^{-6} ⋅ x^{\frac{2}{9}} = x^{-6 + \frac{2}{9}} = x^{-5\frac{7}{9}}\)

\({\sqrt[5]{x^{4}} \over \sqrt[5]{x^{2}}} = {x^{\frac{4}{5}} \over x^{\frac{2}{5}}} = x^{\frac{4}{5} - \frac{2}{5}} = x^{\frac{2}{5}}\)

\(\sqrt[7]{{1 \over p^{5}}} = \sqrt[7]{p^{-5}} = p^{-\frac{5}{7}}\)

\(\sqrt[3]{a^{12}} = a^{\frac{12}{3}} = a^{4}\)

\({p^{4} \over p^{5} ⋅ \sqrt[4]{p^{3}}} = {p^{4} \over p^{5} ⋅ p^{\frac{3}{4}}} = {p^{4} \over p^{5\frac{3}{4}}} = p^{4 - 5\frac{3}{4}} = p^{-1\frac{3}{4}}\)

\({6 \over p^{5}}\)

\({3 b^{9} \over 5 a^{5}}\)

\(({1 \over 4} p)^{-5} = (4^{-1} ⋅ p)^{-5} = (4^{-1})^{-5} ⋅ p^{-5} = 4^{5} ⋅ p^{-5} = {1\,024 \over p^{5}}\)

\(4 x^{6\frac{5}{7}} = 4 ⋅ x^{6} ⋅ x^{\frac{5}{7}} = 4 x^{6} ⋅ \sqrt[7]{x^{5}}\)

\(\frac{2}{3} a^{-\frac{1}{9}} b^{\frac{1}{2}} = \frac{2}{3} ⋅ {1 \over a^{\frac{1}{9}}} ⋅ b^{\frac{1}{2}} = {2 ⋅ \sqrt{b} \over 3 ⋅ \sqrt[9]{a}}\)

\((4 p)^{-5} = 4^{-5} ⋅ p^{-5} = {1 \over 4^{5}} ⋅ {1 \over p^{5}} = {1 \over 1\,024 p^{5}}\)