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

\({7 p^{3} \over 9 p^{5}} = {7 \over 9} ⋅ {p^{3} \over p^{5}} = {7 \over 9} ⋅ p^{3 - 5} = {7 \over 9} p^{-2}\)

\({x^{3} \over x^{-9}} = x^{3 - -9} = x^{12}\)

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

\((x^{6})^{-7} = x^{6 ⋅ -7} = x^{-42}\)

\(p^{2} ⋅ {1 \over p^{5}} = p^{2} ⋅ p^{-5} = p^{2 + -5} = p^{-3}\)

\({({1 \over x^{7}}) \over x^{2}} = {x^{-7} \over x^{2}} = x^{-7 - 2} = x^{-9}\)

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

\({9 a^{3} b \over 2 a b^{3}} = {9 \over 2} ⋅ {a^{3} \over a^{1}} ⋅ {b^{1} \over b^{3}} = {9 \over 2} ⋅ a^{3 - 1} ⋅ a^{1 - 3} = 4\frac{1}{2} a^{2} b^{-2}\)

\({a^{0} \over a^{2}} = a^{0 - 2} = a^{-2}\)

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

\(x^{3} ⋅ \sqrt[8]{x^{5}} = x^{3} ⋅ x^{\frac{5}{8}} = x^{3 + \frac{5}{8}} = x^{3\frac{5}{8}}\)

\({x^{3} \over \sqrt[9]{x^{2}}} = {x^{3} \over x^{\frac{2}{9}}} = x^{3 - \frac{2}{9}} = x^{2\frac{7}{9}}\)

\({1 \over a^{3}} ⋅ \sqrt[9]{a^{7}} = a^{-3} ⋅ a^{\frac{7}{9}} = a^{-3 + \frac{7}{9}} = a^{-2\frac{2}{9}}\)

\({\sqrt[7]{p^{6}} \over \sqrt[7]{p^{2}}} = {p^{\frac{6}{7}} \over p^{\frac{2}{7}}} = p^{\frac{6}{7} - \frac{2}{7}} = p^{\frac{4}{7}}\)

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

\(\sqrt{x^{10}} = x^{\frac{10}{2}} = x^{5}\)

\({a^{4} \over a^{9} ⋅ \sqrt[7]{a^{2}}} = {a^{4} \over a^{9} ⋅ a^{\frac{2}{7}}} = {a^{4} \over a^{9\frac{2}{7}}} = a^{4 - 9\frac{2}{7}} = a^{-5\frac{2}{7}}\)

\({7 \over a^{6}}\)

\({7 b^{4} \over 9 a^{3}}\)

\(({1 \over 2} x)^{-4} = (2^{-1} ⋅ x)^{-4} = (2^{-1})^{-4} ⋅ x^{-4} = 2^{4} ⋅ x^{-4} = {16 \over x^{4}}\)

\(4 a^{8\frac{2}{7}} = 4 ⋅ a^{8} ⋅ a^{\frac{2}{7}} = 4 a^{8} ⋅ \sqrt[7]{a^{2}}\)

\(\frac{1}{5} x^{-\frac{3}{8}} y^{\frac{8}{9}} = \frac{1}{5} ⋅ {1 \over x^{\frac{3}{8}}} ⋅ y^{\frac{8}{9}} = {1 ⋅ \sqrt[9]{y^{8}} \over 5 ⋅ \sqrt[8]{x^{3}}}\)

\((5 x)^{-2} = 5^{-2} ⋅ x^{-2} = {1 \over 5^{2}} ⋅ {1 \over x^{2}} = {1 \over 25 x^{2}}\)