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

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

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

\((x^{8})^{-9} = x^{8 ⋅ -9} = x^{-72}\)

\(p^{5} ⋅ {1 \over p^{9}} = p^{5} ⋅ p^{-9} = p^{5 + -9} = p^{-4}\)

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

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

\({4 \over x^{9}}\)

\({3 x \over 8 x^{6}} = {3 \over 8} ⋅ {x \over x^{6}} = {3 \over 8} ⋅ x^{1 - 6} = {3 \over 8} x^{-5}\)

\({x^{7} \over ({1 \over x^{8}})} = {x^{7} \over x^{-8}} = x^{7 - -8} = x^{15}\)

\({4 p^{6} q \over 3 p^{5} q^{3}} = {4 \over 3} ⋅ {p^{6} \over p^{5}} ⋅ {q^{1} \over q^{3}} = {4 \over 3} ⋅ p^{6 - 5} ⋅ p^{1 - 3} = 1\frac{1}{3} p q^{-2}\)

\(x^{2} ⋅ \sqrt[6]{x} = x^{2} ⋅ x^{\frac{1}{6}} = x^{2 + \frac{1}{6}} = x^{2\frac{1}{6}}\)

\(x^{8} ⋅ \sqrt[5]{x^{4}} = x^{8} ⋅ x^{\frac{4}{5}} = x^{8 + \frac{4}{5}} = x^{8\frac{4}{5}}\)

\({a^{3} \over \sqrt[8]{a^{3}}} = {a^{3} \over a^{\frac{3}{8}}} = a^{3 - \frac{3}{8}} = a^{2\frac{5}{8}}\)

\({1 \over a^{8}} ⋅ \sqrt[5]{a^{3}} = a^{-8} ⋅ a^{\frac{3}{5}} = a^{-8 + \frac{3}{5}} = a^{-7\frac{2}{5}}\)

\({\sqrt[6]{a^{5}} \over \sqrt[8]{a^{3}}} = {a^{\frac{5}{6}} \over a^{\frac{3}{8}}} = a^{\frac{5}{6} - \frac{3}{8}} = a^{\frac{11}{24}}\)

\(\sqrt[8]{{1 \over a^{3}}} = \sqrt[8]{a^{-3}} = a^{-\frac{3}{8}}\)

\(\sqrt[5]{x^{20}} = x^{\frac{20}{5}} = x^{4}\)

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

\({3 q^{8} \over 5 p^{7}}\)

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

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

\(9 x^{2\frac{4}{5}} = 9 ⋅ x^{2} ⋅ x^{\frac{4}{5}} = 9 x^{2} ⋅ \sqrt[5]{x^{4}}\)

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