To solve the expression [tex]\( 9^{-2} \)[/tex], let's break it down step by step:
1. Understand Negative Exponents:
A negative exponent means we take the reciprocal of the base raised to the positive of that exponent. So, [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex].
2. Rewrite [tex]\( 9^{-2} \)[/tex]:
Applying the property of negative exponents, we rewrite [tex]\( 9^{-2} \)[/tex] as:
[tex]\[
9^{-2} = \frac{1}{9^2}
\][/tex]
3. Calculate the Positive Exponent:
Now we compute [tex]\( 9^2 \)[/tex]:
[tex]\[
9^2 = 9 \times 9 = 81
\][/tex]
4. Substitute the Computed Value Back:
Substitute [tex]\( 81 \)[/tex] back into the reciprocal expression:
[tex]\[
9^{-2} = \frac{1}{9^2} = \frac{1}{81}
\][/tex]
Hence, the expression [tex]\( 9^{-2} \)[/tex] is equivalent to [tex]\(\frac{1}{81}\)[/tex].
This means the correct choice from the given options is:
[tex]\[
\boxed{\frac{1}{81}}
\][/tex]
