Answer
$\dfrac{3xy^{3}}{z^{3}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\left( \dfrac{9x^5y^4z^{-7}}{x^3y^{-2}z^{-1}} \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{9x^5y^4z^{-7}}{x^3y^{-2}z^{-1}} \right)^{\frac{1}{2}}
\\\\=
\left( 9x^{5-3}y^{4-(-2)}z^{-7-(-1)} \right)^{\frac{1}{2}}
\\\\=
\left( 9x^{2}y^{6}z^{-6} \right)^{\frac{1}{2}}
\\\\=
\left( 3^2x^{2}y^{6}z^{-6} \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( 3^2x^{2}y^{6}z^{-6} \right)^{\frac{1}{2}}
\\\\=
3^{2\cdot\frac{1}{2}}x^{2\cdot\frac{1}{2}}y^{6\cdot\frac{1}{2}}z^{-6\cdot\frac{1}{2}}
\\\\=
3^{1}x^{1}y^{3}z^{-3}
\\\\=
3xy^{3}z^{-3}
.\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}
3xy^{3}z^{-3}
\\\\=
\dfrac{3xy^{3}}{z^{3}}
.\end{array}
