Answer
$\dfrac{25}{9c^{4}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\left( \dfrac{3c^5d^2}{5c^3d^2} \right)^{-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{3c^5d^2}{5c^3d^2} \right)^{-2}
\\\\
\left( \dfrac{3c^{5-3}d^{2-2}}{5} \right)^{-2}
\\\\
\left( \dfrac{3c^{2}d^{0}}{5} \right)^{-2}
\\\\
\left( \dfrac{3c^{2}(1)}{5} \right)^{-2}
\\\\
\left( \dfrac{3c^{2}}{5} \right)^{-2}
.\end{array}
Using the Power of a Quotient Rule of the laws of exponents, which is given by $\left( \dfrac{x^m}{y^n} \right)^p=\dfrac{x^{mp}}{y^{np}},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\left( \dfrac{3c^{2}}{5} \right)^{-2}
\\\\=
\dfrac{3^{-2}c^{2(-2)}}{5^{-2}}
\\\\=
\dfrac{3^{-2}c^{-4}}{5^{-2}}
.\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{3^{-2}c^{-4}}{5^{-2}}
\\\\=
\dfrac{5^{2}}{3^{2}c^{4}}
\\\\=
\dfrac{25}{9c^{4}}
.\end{array}
