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

$\dfrac{5\sqrt[]{x}}{2}$
Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the given expression is equivalent to \begin{array}{l}\require{cancel} \dfrac{\sqrt[]{75x}}{2\sqrt[]{3}} \\\\= \dfrac{1}{2}\cdot\sqrt[]{\dfrac{75x}{3}} \\\\= \dfrac{1}{2}\cdot\sqrt[]{25x} \\\\= \dfrac{\sqrt[]{25x}}{2} .\end{array} Extracting the factors that are perfect powers of the index, the expression above simplifies to \begin{array}{l}\require{cancel} \dfrac{\sqrt[]{25x}}{2} \\\\= \dfrac{\sqrt[]{(5)^2\cdot x}}{2} \\\\= \dfrac{5\sqrt[]{x}}{2} .\end{array}