$a)\sqrt[5]{7}=7^{\frac{1}{5}}$

$b)\sqrt[3]{7^2}={\left(7^2\right)}^{\frac{1}{3}}=7^{\frac{2}{3}}$

$c)\frac{1}{\sqrt[3]{7}}={\left(\sqrt[3]{7}\right)}^{-1}={\left(7^{\frac{1}{3}}\right)}^{-1}=7^{-\frac{1}{3}}$

$d)\frac{1}{\sqrt[5]{7^3}}={\left(\sqrt[5]{7^3}\right)}^{-1}={\left({\left(7^3\right)}^{\frac{1}{5}}\right)}^{-1}=7^{-\frac{3}{5}}$

$e)7\cdot \sqrt[5]{7}=7\cdot 7^{\frac{1}{5}}=7^{1+\frac{1}{5}}=7^{\frac{6}{5}}$