Answer
$d=\pm \dfrac{\sqrt{kSI}}{I}$
Work Step by Step
Using the properties of equality, in terms of $
d
$, the given equation, $
S=\dfrac{Id^2}{k}
,$ is equivalent to
\begin{align*}\require{cancel}
k(S)&=\left(\dfrac{Id^2}{\cancel k}\right)\cancel k
\\\\
kS&=Id^2
\\\\
\dfrac{kS}{I}&=\dfrac{\cancel Id^2}{\cancel I}
\\\\
\dfrac{kS}{I}&=d^2
\\\\
d^2&=\dfrac{kS}{I}
.\end{align*}
Taking the square root of both sides (Square Root Property), the equation above is equivalent to
\begin{align*}
d&=\pm\sqrt{\dfrac{kS}{I}}
.\end{align*}
Using the properties of radicals, the equation above is equivalent to
\begin{align*}
d&=\pm\sqrt{\dfrac{kS}{I}\cdot\dfrac{I}{I}}
&(\text{rationalize the denominator})
\\\\
d&=\pm\sqrt{\dfrac{1}{I^2}\cdot kSI}
\\\\
d&=\pm\sqrt{\dfrac{1}{I^2}}\cdot \sqrt{kSI}
\\\\
d&=\pm \dfrac{1}{I}\cdot \sqrt{kSI}
\\\\
d&=\pm \dfrac{\sqrt{kSI}}{I}
.\end{align*}
Hence, $
S=\dfrac{Id^2}{k}
$ is equivalent to
\begin{align*}
d=\pm \dfrac{\sqrt{kSI}}{I}
.\end{align*}
