Alright, let's break this down step-by-step.
We start with the expression \(\left(9^{-2}\right)^8\).
To simplify this, we will use the power of a power rule, which states: \((a^m)^n = a^{m \cdot n}\).
1. First, identify the base and the exponents:
- Base: \(9\)
- Exponents: \(-2\) and \(8\)
2. Apply the power of a power rule:
[tex]\[
\left(9^{-2}\right)^8 = 9^{-2 \cdot 8}
\][/tex]
3. Multiply the exponents:
[tex]\[
9^{-2 \cdot 8} = 9^{-16}
\][/tex]
4. Express \(9^{-16}\) in a form that makes it easier to compare with the given options. Negative exponents indicate division (reciprocals), so we recast \(9^{-16}\) as follows:
[tex]\[
9^{-16} = \frac{1}{9^{16}}
\][/tex]
Thus, the expression \(\left(9^{-2}\right)^8\) simplifies to \(\frac{1}{9^{16}}\).
Now, let's identify which option this matches:
A. \(-81^{32}\): This does not match our simplified expression.
B. \(\frac{1}{9^{16}}\): This matches perfectly with our simplified expression.
C. \(\frac{1}{9^{10}}\): This does not match.
D. \(81^8\): This also does not match.
Therefore, the correct answer is:
B [tex]\(\frac{1}{9^{16}}\)[/tex]
