## Intermediate Algebra (12th Edition)

$\dfrac{1-y\sqrt{2y}}{2}$
$\bf{\text{Solution Outline:}}$ 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}