## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$\sqrt[5]{y^2}$
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the given expression is equivalent to \begin{array}{l}\require{cancel} \dfrac{\sqrt{y}}{\sqrt[10]{y}} \\\\= \dfrac{y^{1/2}}{y^{1/10}} .\end{array} Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to \begin{array}{l}\require{cancel} \dfrac{y^{1/2}}{y^{1/10}} \\\\= y^{\frac{1}{2}-\frac{1}{10}} \\\\= y^{\frac{5}{10}-\frac{1}{10}} \\\\= y^{\frac{4}{10}} \\\\= y^{\frac{2}{5}} .\end{array} Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the given expression is equivalent to \begin{array}{l}\require{cancel} y^{\frac{2}{5}} \\\\= \sqrt[5]{y^2} .\end{array}