To solve the given expression [tex]\(\left(2^{-4} \cdot z^{-3} \right)^5\)[/tex], let's break it down step-by-step:
1. Understanding the outer exponent:
We have the expression [tex]\(\left(2^{-4} \cdot z^{-3} \right)^5\)[/tex]. When raising a product to an exponent, you apply the exponent to each factor inside the product separately. Thus, it becomes:
[tex]\[
\left(2^{-4}\right)^5 \cdot \left(z^{-3}\right)^5
\][/tex]
2. Applying the outer exponent separately:
Now, apply the outer exponent 5 to each factor:
- For [tex]\(2^{-4}\)[/tex]:
[tex]\[
\left(2^{-4}\right)^5 = 2^{-4 \cdot 5} = 2^{-20}
\][/tex]
- For [tex]\(z^{-3}\)[/tex]:
[tex]\[
\left(z^{-3}\right)^5 = z^{-3 \cdot 5} = z^{-15}
\][/tex]
3. Combining the results:
Now combine both of these results:
[tex]\[
2^{-20} \cdot z^{-15}
\][/tex]
4. Simplifying the expression:
Recall that a negative exponent means the reciprocal of the base raised to the corresponding positive exponent. Thus:
[tex]\[
2^{-20} = \frac{1}{2^{20}}
\][/tex]
and
[tex]\[
z^{-15} = \frac{1}{z^{15}}
\][/tex]
Therefore,
[tex]\[
2^{-20} \cdot z^{-15} = \frac{1}{2^{20}} \cdot \frac{1}{z^{15}} = \frac{1}{2^{20} \cdot z^{15}}
\][/tex]
So, the equivalent expression is:
[tex]\[
\frac{1}{2^{20} \cdot z^{15}}
\][/tex]
Accordingly, the correct answer is:
[tex]\[ \boxed{\frac{1}{2^{20} \cdot z^{15}}} \][/tex]
Thus, the correct choice from the given options is:
(A) [tex]\(\frac{1}{2^{20} \cdot z^{15}}\)[/tex]
