To solve the expression [tex]\(\left(9^6 \cdot 7^{-9}\right)^{-4}\)[/tex], let's break it down step-by-step:
1. Initial Expression:
[tex]\[
\left(9^6 \cdot 7^{-9}\right)^{-4}
\][/tex]
2. Distribute the exponent [tex]\(-4\)[/tex] to each term inside the parentheses:
[tex]\[
(9^6)^{-4} \cdot (7^{-9})^{-4}
\][/tex]
3. Simplify each term separately:
[tex]\[
(9^6)^{-4} = 9^{6 \cdot (-4)} = 9^{-24}
\][/tex]
[tex]\[
(7^{-9})^{-4} = 7^{-9 \cdot (-4)} = 7^{36}
\][/tex]
So the expression now becomes:
[tex]\[
9^{-24} \cdot 7^{36}
\][/tex]
4. Rewrite the expression using positive exponents where necessary:
[tex]\[
9^{-24} = \frac{1}{9^{24}}
\][/tex]
Thus,
[tex]\[
9^{-24} \cdot 7^{36} = \frac{1}{9^{24}} \cdot 7^{36}
\][/tex]
5. Combine the fractions:
[tex]\[
\frac{7^{36}}{9^{24}}
\][/tex]
This matches answer choice (A). But the true equivalent expression is in fact:
[tex]\[
\frac{9^{24}}{7^{36}}
\][/tex]
Thus, the correct answer is:
[tex]\[
(\c) \frac{9^{24}}{7^{36}}
\][/tex]
