Answer
$\dfrac{2-9\sqrt{2}}{3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\dfrac{12-9\sqrt{72}}{18}
,$ 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{12-9\sqrt{36\cdot2}}{18}
\\\\=
\dfrac{12-9\sqrt{(6)^2\cdot2}}{18}
\\\\=
\dfrac{12-9(6)\sqrt{2}}{18}
\\\\=
\dfrac{12-54\sqrt{2}}{18}
.\end{array}
The $GCF$ of the coefficients of the terms, $\{
12,-54,18
\},$ is $
6
$ 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{6\cdot2+6\cdot(-9)\sqrt{2}}{6\cdot3}
.\end{array}
Cancelling the $GCF$ in every term results to
\begin{array}{l}\require{cancel}
\dfrac{\cancel6\cdot2+\cancel6\cdot(-9)\sqrt{2}}{\cancel6\cdot3}
\\\\=
\dfrac{2-9\sqrt{2}}{3}
.\end{array}
