Answer
$-\dfrac{5m\sqrt{3mp}}{p}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
-\sqrt{\dfrac{75m^3}{p}}
,$ multiply the numerator and the denominator by an expression equal to $1$ which will make the denominator a perfect power of the index.
$\bf{\text{Solution Details:}}$
Multiplying the numerator and the denominator 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{75m^3}{p}\cdot\dfrac{p}{p}}
\\\\=
-\sqrt{\dfrac{75m^3p}{p^2}}
.\end{array}
Rewriting the radicand using an expression that contains a factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
-\sqrt{\dfrac{25m^2}{p^2}\cdot3mp}
\\\\=
-\sqrt{\left(\dfrac{5m}{p}\right)^2\cdot3mp}
.\end{array}
Extracting the root of the radicand that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
-\dfrac{5m}{p}\sqrt{3mp}
\\\\=
-\dfrac{5m\sqrt{3mp}}{p}
.\end{array}
