Answer
$\dfrac{15}{22h^{2}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\left( \dfrac{3g^5h^{-7}}{4g^2h^3} \right)\left( \dfrac{10g^{-2}h^6}{11gh^{-2}} \right)
,$ 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{3g^5h^{-7}}{4g^2h^3} \right)\left( \dfrac{10g^{-2}h^6}{11gh^{-2}} \right)
\\\\
\left( \dfrac{3g^{5-2}h^{-7-3}}{4} \right)\left( \dfrac{10g^{-2-1}h^{6-(-2)}}{11} \right)
\\\\
\left( \dfrac{3g^{3}h^{-10}}{4} \right)\left( \dfrac{10g^{-3}h^{8}}{11} \right)
.\end{array}
Using the Product Rule of the laws of exponents, which is given by $x^m\cdot x^n=x^{m+n},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\left( \dfrac{3g^{3}h^{-10}}{4} \right)\left( \dfrac{10g^{-3}h^{8}}{11} \right)
\\\\=
\dfrac{3(10)g^{3+(-3)}h^{-10+8}}{4(11)}
\\\\=
\dfrac{3(\cancel{10}^5)g^{0}h^{-2}}{\cancel{4}^2(11)}
\\\\=
\dfrac{15g^{0}h^{-2}}{22}
.\end{array}
Since any expression (except $0$) raised to the $0$ power is $1,$ then the expression above simplifies to
\begin{array}{l}\require{cancel}
\dfrac{15g^{0}h^{-2}}{22}
\\\\
\dfrac{15(1)h^{-2}}{22}
\\\\
\dfrac{15h^{-2}}{22}
.\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}
\dfrac{15h^{-2}}{22}
\\\\=
\dfrac{15}{22h^{2}}
.\end{array}