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

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

\({a^{9} \over a^{-4}} = a^{9 - -4} = a^{13}\)

\(x^{4} ⋅ x^{-5} = x^{4 + -5} = x^{-1}\)

\((p^{8})^{-3} = p^{8 ⋅ -3} = p^{-24}\)

\(x^{5} ⋅ {1 \over x^{8}} = x^{5} ⋅ x^{-8} = x^{5 + -8} = x^{-3}\)

\({({1 \over a^{8}}) \over a^{4}} = {a^{-8} \over a^{4}} = a^{-8 - 4} = a^{-12}\)

\({p^{6} \over ({1 \over p^{8}})} = {p^{6} \over p^{-8}} = p^{6 - -8} = p^{14}\)

\({8 x^{3} y^{5} \over 7 x y^{8}} = {8 \over 7} ⋅ {x^{3} \over x^{1}} ⋅ {y^{5} \over y^{8}} = {8 \over 7} ⋅ x^{3 - 1} ⋅ x^{5 - 8} = 1\frac{1}{7} x^{2} y^{-3}\)

\({a^{0} \over a^{4}} = a^{0 - 4} = a^{-4}\)

\(x^{3} ⋅ \sqrt[7]{x} = x^{3} ⋅ x^{\frac{1}{7}} = x^{3 + \frac{1}{7}} = x^{3\frac{1}{7}}\)

\(a^{3} ⋅ \sqrt[5]{a^{3}} = a^{3} ⋅ a^{\frac{3}{5}} = a^{3 + \frac{3}{5}} = a^{3\frac{3}{5}}\)

\({a^{7} \over \sqrt[7]{a^{2}}} = {a^{7} \over a^{\frac{2}{7}}} = a^{7 - \frac{2}{7}} = a^{6\frac{5}{7}}\)

\({1 \over x^{4}} ⋅ \sqrt[7]{x^{6}} = x^{-4} ⋅ x^{\frac{6}{7}} = x^{-4 + \frac{6}{7}} = x^{-3\frac{1}{7}}\)

\({\sqrt[6]{p^{5}} \over \sqrt[3]{p^{2}}} = {p^{\frac{5}{6}} \over p^{\frac{2}{3}}} = p^{\frac{5}{6} - \frac{2}{3}} = p^{\frac{1}{6}}\)

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

\(\sqrt{x^{4}} = x^{\frac{4}{2}} = x^{2}\)

\({a^{4} \over a^{5} ⋅ \sqrt[9]{a^{7}}} = {a^{4} \over a^{5} ⋅ a^{\frac{7}{9}}} = {a^{4} \over a^{5\frac{7}{9}}} = a^{4 - 5\frac{7}{9}} = a^{-1\frac{7}{9}}\)

\({8 \over a^{3}}\)

\({8 y^{2} \over 9 x^{3}}\)

\(({1 \over 3} p)^{-2} = (3^{-1} ⋅ p)^{-2} = (3^{-1})^{-2} ⋅ p^{-2} = 3^{2} ⋅ p^{-2} = {9 \over p^{2}}\)

\(8 x^{5\frac{7}{9}} = 8 ⋅ x^{5} ⋅ x^{\frac{7}{9}} = 8 x^{5} ⋅ \sqrt[9]{x^{7}}\)

\(\frac{4}{7} a^{-\frac{1}{5}} b^{\frac{1}{9}} = \frac{4}{7} ⋅ {1 \over a^{\frac{1}{5}}} ⋅ b^{\frac{1}{9}} = {4 ⋅ \sqrt[9]{b} \over 7 ⋅ \sqrt[5]{a}}\)

\((2 x)^{-3} = 2^{-3} ⋅ x^{-3} = {1 \over 2^{3}} ⋅ {1 \over x^{3}} = {1 \over 8 x^{3}}\)