Answer
$\dfrac{4-2\sqrt{2}}{3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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}
