Answer
$e=\pm\dfrac{\sqrt{2Ekr}}{k}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the properties of equality to solve the given equation, $
E=\dfrac{e^2k}{2r}
,$ for $
e
.$
$\bf{\text{Solution Details:}}$
Multiplying both sides by $
2r
,$ and then dividing by $
k
,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
2Er=e^2k
\\\\
\dfrac{2Er}{k}=e^2
\\\\
e^2=\dfrac{2Er}{k}
.\end{array}
Taking the square root of both sides (Square Root Principle) results to
\begin{array}{l}\require{cancel}
e=\pm\sqrt{\dfrac{2Er}{k}}
.\end{array}
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{2Er}{k}\cdot\dfrac{k}{k}}
\\\\
e=\pm\sqrt{\dfrac{2Ekr}{k^2}}
\\\\
e=\pm\sqrt{\dfrac{1}{k^2}\cdot2Ekr}
\\\\
e=\pm\sqrt{\left( \dfrac{1}{k}\right)^2\cdot2Ekr}
\\\\
e=\pm\left| \dfrac{1}{k}\right|\sqrt{2Ekr}
.\end{array}
Assuming that all variables are positive, the equation above is equivalent to
\begin{array}{l}\require{cancel}
e=\pm\dfrac{1}{k}\sqrt{2Ekr}
\\\\
e=\pm\dfrac{\sqrt{2Ekr}}{k}
.\end{array}
