#### Answer

$\dfrac{1-y\sqrt{2y}}{2}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\dfrac{11y-\sqrt{242y^5}}{22y}
,$ 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{11y-\sqrt{121y^4\cdot 2y}}{22y}
\\\\=
\dfrac{11y-\sqrt{(11y^2)^2\cdot 2y}}{22y}
\\\\=
\dfrac{11y-11y^2\sqrt{2y}}{22y}
.\end{array}
The $GCF$ of the constants of the terms $\{
11,-11,22
\}$ is $
11
$ since it is the highest number that can divide all the given constants. The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variable/s $\{
y,y^2,y
\}$ is $
y
.$ Hence, the entire expression has $GCF=
11y
.$ Writing the given expression as factors using the $GCF$ results to
\begin{array}{l}\require{cancel}
\dfrac{11y\cdot1-11y\cdot y\sqrt{2y}}{11y\cdot2}
.\end{array}
Cancelling the $GCF$ in every term results to
\begin{array}{l}\require{cancel}
\dfrac{\cancel{11y}\cdot1-\cancel{11y}\cdot y\sqrt{2y}}{\cancel{11y}\cdot2}
\\\\=
\dfrac{1-y\sqrt{2y}}{2}
.\end{array}