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

$30a^{5}\sqrt{3}$
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression, $3\sqrt{5a^7}\cdot2\sqrt{15a^3} ,$ is equivalent to \begin{array}{l}\require{cancel} 3\sqrt{5a^7}\cdot2\sqrt{15a^3} \\\\= 3(2)\sqrt{5a^7(15a^3)} \\\\= 6\sqrt{75a^{10}} .\end{array} Extracting the factor that is a perfect power of the index (all radicands are assumed positive), then \begin{array}{l}\require{cancel} 6\sqrt{75a^{10}} \\\\= 6\sqrt{25a^{10}\cdot3} \\\\= 6\sqrt{(5a^{5})^2\cdot3} \\\\= 6\cdot5a^{5}\sqrt{3} \\\\= 30a^{5}\sqrt{3} .\end{array}