Answered

1. Which expression is equivalent to [tex]\left(9^{-2}\right)^8[/tex]?

A. [tex]-81^{32}[/tex]
B. [tex]\frac{1}{9^{16}}[/tex]
C. [tex]\frac{1}{9^{10}}[/tex]
D. [tex]81^8[/tex]



Answer :

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]