Answer
$\dfrac{y}{x}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\left( \dfrac{x^{-2}y^{-1/3}}{x^{-5/3}y^{-2/3}} \right)^3
,$ use the laws of exponents.
$\bf{\text{Solution Details:}}$
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{x^{-2(3)}y^{-\frac{1}{3}\cdot3}}{x^{-\frac{5}{3}\cdot3}y^{-\frac{2}{3}\cdot3}}
\\\\=
\dfrac{x^{-6}y^{-1}}{x^{-5}y^{-2}}
.\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}
x^{-6-(-5)}y^{-1-(-2)}
\\\\=
x^{-6+5}y^{-1+2}
\\\\=
x^{-1}y^{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}
\dfrac{y^{1}}{x^{1}}
\\\\=
\dfrac{y}{x}
.\end{array}
