Answer
$\dfrac{1}{p^2q^4}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the laws of exponents to simplify the given expression, $
\dfrac{(p^{15}q^{12})^{-4/3}}{(p^{24}q^{16})^{-3/4}}
.$
$\bf{\text{Solution Details:}}$
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}
\dfrac{p^{15\left(-\frac{4}{3} \right)}q^{12\left(-\frac{4}{3} \right)}}{p^{24\left(-\frac{3}{4} \right)}q^{16\left(-\frac{3}{4} \right)}}
\\\\=
\dfrac{p^{-20}q^{-16}}{p^{-18}q^{-12}}
.\end{array}
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}
p^{-20-(-18)}q^{-16-(-12)}
\\\\=
p^{-20+18}q^{-16+12}
\\\\=
p^{-2}q^{-4}
.\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{1}{p^2q^4}
.\end{array}
