To determine which expression is equivalent to [tex]\((3a)^{-2}\)[/tex], we will follow a detailed, step-by-step approach.
### Step-by-Step Solution
1. Understanding the Negative Exponent:
The expression [tex]\((3a)^{-2}\)[/tex] involves a negative exponent. The property of negative exponents states that [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex]. Applying this property to our expression:
[tex]\[
(3a)^{-2} = \frac{1}{(3a)^2}
\][/tex]
2. Simplifying the Denominator:
We now need to simplify [tex]\((3a)^2\)[/tex]. This involves applying the exponent to both the coefficient (3) and the variable (a):
[tex]\[
(3a)^2 = (3)^2 \cdot (a)^2
\][/tex]
3. Calculating Each Component:
- The square of 3 is:
[tex]\[
3^2 = 9
\][/tex]
- The square of [tex]\(a\)[/tex] is:
[tex]\[
a^2
\][/tex]
Therefore:
[tex]\[
(3a)^2 = 9a^2
\][/tex]
4. Putting it all together:
Substituting back into our fraction, we have:
[tex]\[
(3a)^{-2} = \frac{1}{(3a)^2} = \frac{1}{9a^2}
\][/tex]
### Summary
The expression [tex]\((3a)^{-2}\)[/tex] simplifies to [tex]\(\frac{1}{9a^2}\)[/tex]. Hence, the equivalent expression is:
[tex]\[
\boxed{\frac{1}{9a^2}}
\][/tex]
Thus, the correct answer is the first option:
[tex]\[
\frac{1}{9 a^2}
\][/tex]
