Intermediate Algebra (12th Edition)

$-3\sqrt{2k}$
$\bf{\text{Solution Outline:}}$ To simplify the given radical expression, $\sqrt{18k}-\sqrt{72k} ,$ simplify first each term by expressing the radicand as a factor that is a perfect power of the index. Then, extract the root. Finally, combine the like radicals. $\bf{\text{Solution Details:}}$ Expressing the radicand as an expression that contains a factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} \sqrt{9\cdot2k}-\sqrt{36\cdot2k} \\\\= \sqrt{(3)^2\cdot2k}-\sqrt{(6)^2\cdot2k} .\end{array} Extracting the roots of the factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} 3\sqrt{2k}-6\sqrt{2k} .\end{array} By combining like radicals, the expression above is equivalent to \begin{array}{l}\require{cancel} (3-6)\sqrt{2k} \\\\= -3\sqrt{2k} .\end{array}