Answer
$\dfrac{c^{11/3}}{b^{11/4}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the laws of exponents to simplify the given expression, $
\left( \dfrac{b^{-3/2}}{c^{-5/3}} \right)^2 \left( b^{-1/4}c^{-1/3} \right)^{-1}
.$
$\bf{\text{Solution Details:}}$
Using the extended Power Rule of the laws of exponents which states that $\left( \dfrac{x^my^n}{z^p} \right)^q=\dfrac{x^{mq}y^{nq}}{z^{pq}},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\left( \dfrac{b^{-\frac{3}{2}\cdot2}}{c^{-\frac{5}{3}\cdot2}} \right) b^{-\frac{1}{4}\cdot(-1)}c^{-\frac{1}{3}\cdot(-1)}
\\\\=
\left( \dfrac{b^{-3}}{c^{-\frac{10}{3}}} \right) b^{\frac{1}{4}}c^{\frac{1}{3}}
\\\\=
\dfrac{b^{-3}b^{\frac{1}{4}}c^{\frac{1}{3}}}{c^{-\frac{10}{3}}}
.\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}
\dfrac{b^{-3+\frac{1}{4}}c^{\frac{1}{3}}}{c^{-\frac{10}{3}}}
\\\\=
\dfrac{b^{-\frac{12}{4}+\frac{1}{4}}c^{\frac{1}{3}}}{c^{-\frac{10}{3}}}
\\\\=
\dfrac{b^{-\frac{11}{4}}c^{\frac{1}{3}}}{c^{-\frac{10}{3}}}
.\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}
b^{-\frac{11}{4}}c^{\frac{1}{3}-\left( -\frac{10}{3} \right)}
\\\\=
b^{-\frac{11}{4}}c^{\frac{1}{3}+\frac{10}{3}}
\\\\=
b^{-\frac{11}{4}}c^{\frac{11}{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}
\dfrac{c^{\frac{11}{3}}}{b^{\frac{11}{4}}}
\\\\=
\dfrac{c^{11/3}}{b^{11/4}}
.\end{array}