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

\({4 x^{3} \over 5 x^{5}} = {4 \over 5} ⋅ {x^{3} \over x^{5}} = {4 \over 5} ⋅ x^{3 - 5} = {4 \over 5} x^{-2}\)

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

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

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

\(p^{8} ⋅ {1 \over p^{9}} = p^{8} ⋅ p^{-9} = p^{8 + -9} = p^{-1}\)

\({({1 \over x^{9}}) \over x^{5}} = {x^{-9} \over x^{5}} = x^{-9 - 5} = x^{-14}\)

\({a^{2} \over ({1 \over a^{3}})} = {a^{2} \over a^{-3}} = a^{2 - -3} = a^{5}\)

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

\({a^{3} \over a^{0}} = a^{3 - 0} = a^{3}\)

\(a^{8} ⋅ \sqrt[7]{a} = a^{8} ⋅ a^{\frac{1}{7}} = a^{8 + \frac{1}{7}} = a^{8\frac{1}{7}}\)

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

\({p^{3} \over \sqrt[7]{p^{4}}} = {p^{3} \over p^{\frac{4}{7}}} = p^{3 - \frac{4}{7}} = p^{2\frac{3}{7}}\)

\({1 \over x^{5}} ⋅ \sqrt[9]{x^{7}} = x^{-5} ⋅ x^{\frac{7}{9}} = x^{-5 + \frac{7}{9}} = x^{-4\frac{2}{9}}\)

\({\sqrt[8]{a^{7}} \over \sqrt[3]{a^{2}}} = {a^{\frac{7}{8}} \over a^{\frac{2}{3}}} = a^{\frac{7}{8} - \frac{2}{3}} = a^{\frac{5}{24}}\)

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

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

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

\({7 \over x^{2}}\)

\({4 b^{9} \over 5 a^{8}}\)

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

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

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

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