To simplify the expression [tex]\(\left(5^{-2} \cdot 4^{-4}\right)^{-2}\)[/tex], let's follow a step-by-step approach.
### Step 1: Distribute the Outer Exponent
The given expression is [tex]\(\left(5^{-2} \cdot 4^{-4}\right)^{-2}\)[/tex].
We need to distribute the outer exponent [tex]\(-2\)[/tex] to each base inside the parentheses:
[tex]\[
\left(5^{-2}\right)^{-2} \cdot \left(4^{-4}\right)^{-2}
\][/tex]
### Step 2: Apply the Power Rule
When raising a power to another power, you multiply the exponents. Therefore, we apply this rule to both terms:
[tex]\[
(5^{-2})^{-2} = 5^{(-2) \cdot (-2)} = 5^{4}
\][/tex]
[tex]\[
(4^{-4})^{-2} = 4^{(-4) \cdot (-2)} = 4^{8}
\][/tex]
### Step 3: Combine the Results
Now, we combine these simplified terms:
[tex]\[
5^4 \cdot 4^8
\][/tex]
Thus, the expression [tex]\(\left(5^{-2} \cdot 4^{-4}\right)^{-2}\)[/tex] simplifies to:
[tex]\[
\boxed{5^4 \cdot 4^8}
\][/tex]
So, the correct answer is:
A. [tex]\(5^4 \cdot 4^8\)[/tex]
