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

\({x^{7} \over x^{-4}} = x^{7 - -4} = x^{11}\)

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

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

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

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

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

\({8 \over a^{4}}\)

\({3 a^{4} \over 5 a^{7}} = {3 \over 5} ⋅ {a^{4} \over a^{7}} = {3 \over 5} ⋅ a^{4 - 7} = {3 \over 5} a^{-3}\)

\({x^{2} \over ({1 \over x^{8}})} = {x^{2} \over x^{-8}} = x^{2 - -8} = x^{10}\)

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

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

\(p^{8} ⋅ \sqrt[3]{p^{2}} = p^{8} ⋅ p^{\frac{2}{3}} = p^{8 + \frac{2}{3}} = p^{8\frac{2}{3}}\)

\({x^{8} \over \sqrt[9]{x^{8}}} = {x^{8} \over x^{\frac{8}{9}}} = x^{8 - \frac{8}{9}} = x^{7\frac{1}{9}}\)

\({1 \over x^{9}} ⋅ \sqrt[9]{x^{5}} = x^{-9} ⋅ x^{\frac{5}{9}} = x^{-9 + \frac{5}{9}} = x^{-8\frac{4}{9}}\)

\({\sqrt[8]{x^{5}} \over \sqrt[7]{x^{4}}} = {x^{\frac{5}{8}} \over x^{\frac{4}{7}}} = x^{\frac{5}{8} - \frac{4}{7}} = x^{\frac{3}{56}}\)

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

\(\sqrt{x^{8}} = x^{\frac{8}{2}} = x^{4}\)

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

\({2 b^{7} \over 3 a^{3}}\)

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

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

\(4 a^{8\frac{2}{3}} = 4 ⋅ a^{8} ⋅ a^{\frac{2}{3}} = 4 a^{8} ⋅ \sqrt[3]{a^{2}}\)

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