To determine which expression is equivalent to [tex]\( 9^{-2} \)[/tex], we need to recall the rules of exponents. Specifically, when dealing with a negative exponent, the expression [tex]\( a^{-b} \)[/tex] is equivalent to [tex]\( \frac{1}{a^b} \)[/tex].
Given [tex]\( 9^{-2} \)[/tex]:
1. Convert the negative exponent to a reciprocal format:
[tex]\[
9^{-2} = \frac{1}{9^2}
\][/tex]
2. Calculate [tex]\( 9^2 \)[/tex]:
[tex]\[
9^2 = 9 \times 9 = 81
\][/tex]
3. Place the result in the reciprocal expression:
[tex]\[
9^{-2} = \frac{1}{81}
\][/tex]
Therefore, the expression [tex]\( \frac{1}{81} \)[/tex] is equivalent to [tex]\( 9^{-2} \)[/tex].
The correct answer is:
[tex]\(\frac{1}{81}\)[/tex]
