Answer
$\dfrac{4mn^{4}}{p}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\left( \dfrac{16m^3n^6p}{mn^{-2}p^3} \right)^{\frac{1}{2}}
,$ use the laws of exponents.
$\bf{\text{Solution Details:}}$
Using the Quotient Rule of the laws of exponents, which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to
\begin{array}{l}\require{cancel}
\left( \dfrac{16m^3n^6p}{mn^{-2}p^3} \right)^{\frac{1}{2}}
\\\\=
\left( 16m^{3-1}n^{6-(-2)}p^{1-3} \right)^{\frac{1}{2}}
\\\\=
\left( 16m^{2}n^{8}p^{-2} \right)^{\frac{1}{2}}
.\end{array}
Using the extended Power Rule of the laws of exponents, which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\left( 16m^{2}n^{8}p^{-2} \right)^{\frac{1}{2}}
\\\\=
\left( 4^2m^{2}n^{8}p^{-2} \right)^{\frac{1}{2}}
\\\\=
4^{2\cdot\frac{1}{2}}m^{2\cdot\frac{1}{2}}n^{8\cdot\frac{1}{2}}p^{-2\cdot\frac{1}{2}}
\\\\=
4^{1}m^{1}n^{4}p^{-1}
\\\\=
4mn^{4}p^{-1}
.\end{array}
Using the Negative Exponent Rule of the laws of exponents, which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
4mn^{4}p^{-1}
\\\\=
\dfrac{4mn^{4}}{p^1}
\\\\=
\dfrac{4mn^{4}}{p}
.\end{array}