The expression that is correctly developed to use the Product Rule of Exponents is:
\((5^{2})^{9}\)
This is because the Product Rule of Exponents states that when you raise a power to a power, you multiply the exponents. In this case, you would calculate it as \(5^{2 \cdot 9} = 5^{18}\).
The Product Rule itself generally refers to the rule that states when multiplying two powers with the same base, you add the exponents, for example: \(a^{m} \cdot a^{n} = a^{m+n}\).
However, the expression \((5^{2})^{9}\) correctly relates to the rules of exponentiation. In contrast, the other expressions either do not require the Product Rule or aren't correctly using it.