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