## Intermediate Algebra (12th Edition)

$\dfrac{4-2\sqrt{2}}{3}$
$\bf{\text{Solution Outline:}}$ To simplify the the given expression, $\dfrac{16-4\sqrt{8}}{12} ,$ simplify the radicand that contains a factor that is a perfect power of the index Then, find the $GCF$ of all the terms and express all terms as factors using the $GCF.$ Finally, cancel the $GCF$ in all the terms. $\bf{\text{Solution Details:}}$ Writing the radicand as an expression containing a factor that is a perfect power of the index and extracting the root of that factor result to \begin{array}{l}\require{cancel} \dfrac{16-4\sqrt{4\cdot2}}{12} \\\\= \dfrac{16-4\sqrt{(2)^2\cdot2}}{12} \\\\= \dfrac{16-4(2)\sqrt{2}}{12} \\\\= \dfrac{16-8\sqrt{2}}{12} .\end{array} The $GCF$ of the coefficients of the terms, $\{ 16,-8,12 \},$ is $4$ since it is the highest number that can divide all the given coefficients. Writing the given expression as factors using the $GCF$ results to \begin{array}{l}\require{cancel} \dfrac{4\cdot4+4\cdot(-2)\sqrt{2}}{4\cdot3} .\end{array} Cancelling the $GCF$ in every term results to \begin{array}{l}\require{cancel} \dfrac{\cancel{4}\cdot4+\cancel{4}\cdot(-2)\sqrt{2}}{\cancel{4}\cdot3} \\\\= \dfrac{4-2\sqrt{2}}{3} .\end{array}