To simplify the given expression [tex]\(\left(3^{-8} \cdot 7^3\right)^{-2}\)[/tex], let's proceed step-by-step:
1. Apply the outer exponent [tex]\(-2\)[/tex] to each term inside the parentheses:
[tex]\[
\left(3^{-8} \cdot 7^3\right)^{-2}
\][/tex]
This can be rewritten as applying the exponent to each individual part:
[tex]\[
(3^{-8})^{-2} \cdot (7^3)^{-2}
\][/tex]
2. Simplify each term separately:
[tex]\[
(3^{-8})^{-2} \quad \text{and} \quad (7^3)^{-2}
\][/tex]
3. Use the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(3^{-8})^{-2} = 3^{-8 \cdot (-2)} = 3^{16}
\][/tex]
[tex]\[
(7^3)^{-2} = 7^{3 \cdot (-2)} = 7^{-6}
\][/tex]
4. Combine the results:
[tex]\[
(3^{-8} \cdot 7^3)^{-2} = 3^{16} \cdot 7^{-6}
\][/tex]
Comparing the simplified expression [tex]\(3^{16} \cdot 7^{-6}\)[/tex] with the given choices:
- (A) [tex]\(21^{10}\)[/tex]
- (B) [tex]\(\frac{7^6}{3^{16}}\)[/tex]
- (C) [tex]\(3^{16} \cdot 7^{-6}\)[/tex]
The correct answer is:
(C) [tex]\(3^{16} \cdot 7^{-6}\)[/tex]
