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

\({a^{5} \over a^{-2}} = a^{5 - -2} = a^{7}\)

\(x^{2} ⋅ x^{-6} = x^{2 + -6} = x^{-4}\)

\((x^{2})^{-7} = x^{2 ⋅ -7} = x^{-14}\)

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

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

\({x^{0} \over x^{5}} = x^{0 - 5} = x^{-5}\)

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

\({6 x^{4} \over 7 x^{6}} = {6 \over 7} ⋅ {x^{4} \over x^{6}} = {6 \over 7} ⋅ x^{4 - 6} = {6 \over 7} x^{-2}\)

\({p^{3} \over ({1 \over p^{8}})} = {p^{3} \over p^{-8}} = p^{3 - -8} = p^{11}\)

\({5 a^{3} b^{4} \over 2 a^{2} b^{7}} = {5 \over 2} ⋅ {a^{3} \over a^{2}} ⋅ {b^{4} \over b^{7}} = {5 \over 2} ⋅ a^{3 - 2} ⋅ a^{4 - 7} = 2\frac{1}{2} a b^{-3}\)

\(x^{9} ⋅ \sqrt[8]{x} = x^{9} ⋅ x^{\frac{1}{8}} = x^{9 + \frac{1}{8}} = x^{9\frac{1}{8}}\)

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

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

\({1 \over p^{7}} ⋅ \sqrt[8]{p^{3}} = p^{-7} ⋅ p^{\frac{3}{8}} = p^{-7 + \frac{3}{8}} = p^{-6\frac{5}{8}}\)

\({\sqrt[4]{x^{3}} \over \sqrt[8]{x^{5}}} = {x^{\frac{3}{4}} \over x^{\frac{5}{8}}} = x^{\frac{3}{4} - \frac{5}{8}} = x^{\frac{1}{8}}\)

\(\sqrt[9]{{1 \over p^{4}}} = \sqrt[9]{p^{-4}} = p^{-\frac{4}{9}}\)

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

\({x^{3} \over x^{8} ⋅ \sqrt[9]{x^{4}}} = {x^{3} \over x^{8} ⋅ x^{\frac{4}{9}}} = {x^{3} \over x^{8\frac{4}{9}}} = x^{3 - 8\frac{4}{9}} = x^{-5\frac{4}{9}}\)

\({7 y^{5} \over 8 x^{3}}\)

\((5 a)^{-3} = 5^{-3} ⋅ a^{-3} = {1 \over 5^{3}} ⋅ {1 \over a^{3}} = {1 \over 125 a^{3}}\)

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

\(9 a^{4\frac{8}{9}} = 9 ⋅ a^{4} ⋅ a^{\frac{8}{9}} = 9 a^{4} ⋅ \sqrt[9]{a^{8}}\)

\(\frac{3}{8} p^{-\frac{3}{4}} q^{\frac{8}{9}} = \frac{3}{8} ⋅ {1 \over p^{\frac{3}{4}}} ⋅ q^{\frac{8}{9}} = {3 ⋅ \sqrt[9]{q^{8}} \over 8 ⋅ \sqrt[4]{p^{3}}}\)