To determine the expression equivalent to [tex]\((3a)^{-2}\)[/tex], we need to understand the properties of exponents. Specifically, we will use the rule that states [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex].
Let's break it down step-by-step:
1. Start with the given expression:
[tex]\[
(3a)^{-2}
\][/tex]
2. Apply the negative exponent rule [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex]:
[tex]\[
(3a)^{-2} = \frac{1}{(3a)^2}
\][/tex]
3. Now, simplify the denominator [tex]\((3a)^2\)[/tex]. This is the square of the product [tex]\(3a\)[/tex]:
[tex]\[
(3a)^2 = 3^2 \cdot a^2
\][/tex]
4. Compute [tex]\(3^2\)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
5. Thus, the expression in the denominator becomes:
[tex]\[
(3a)^2 = 9a^2
\][/tex]
6. Substitute [tex]\(9a^2\)[/tex] back into the fraction:
[tex]\[
\frac{1}{(3a)^2} = \frac{1}{9a^2}
\][/tex]
Therefore, the expression equivalent to [tex]\((3a)^{-2}\)[/tex] is:
[tex]\[
\frac{1}{9a^2}
\][/tex]
This matches the first option.
Hence, the correct choice is:
[tex]\[
\boxed{\frac{1}{9a^2}}
\][/tex]
