## Intermediate Algebra (12th Edition)

$\dfrac{2\sqrt{13}}{y}$
$\bf{\text{Solution Outline:}}$ To rationalize the given radical expression, $\sqrt{\dfrac{52}{y}} ,$ multiply both the numerator and the denominator by an expression that will make the denominator a perfect power of the index. Note that the variables are assumed to represent positive real numbers. $\bf{\text{Solution Details:}}$ Multiplying the radicand by an expression equal to $1$ which will make the denominator a perfect power of the index results to \begin{array}{l}\require{cancel} \sqrt{\dfrac{52}{y}\cdot\dfrac{y}{y}} \\\\= \sqrt{\dfrac{52y}{y^2}} .\end{array} Writing the radicand as an expression that contains a factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} \sqrt{\dfrac{4}{y^2}\cdot13} \\\\= \sqrt{\left(\dfrac{2}{y}\right)^2\cdot13} .\end{array} Extracting the root of the factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} \dfrac{2}{y}\sqrt{13} \\\\= \dfrac{2\sqrt{13}}{y} .\end{array}