## Intermediate Algebra (12th Edition)

$e=\pm\dfrac{\sqrt{6S}}{6}$
$\bf{\text{Solution Outline:}}$ To solve the given equation, $S=6e^2 ,$ in terms of $e ,$ use the properties of equality and the Square Root Principle to isolate the variable. $\bf{\text{Solution Details:}}$ Using the properties of equality, the equation above is equivalent to \begin{array}{l}\require{cancel} \dfrac{S}{6}=e^2 \\\\ e^2=\dfrac{S}{6} .\end{array} Taking the square root of both sides (Square Root Principle), the equation above is equivalent to \begin{array}{l}\require{cancel} e=\pm\sqrt{\dfrac{S}{6}} .\end{array} Rationalizing the denominator by 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} e=\pm\sqrt{\dfrac{S}{6}\cdot\dfrac{6}{6}} \\\\ e=\pm\sqrt{\dfrac{6S}{36}} \\\\ e=\pm\sqrt{\dfrac{1}{36}\cdot6S} \\\\ e=\pm\sqrt{\left(\dfrac{1}{6}\right)^2\cdot6S} \\\\ e=\pm\dfrac{1}{6}\sqrt{6S} \\\\ e=\pm\dfrac{\sqrt{6S}}{6} .\end{array}