Sure, let's solve the given mathematical expression step by step to find the equivalent expression.
We start with the expression:
[tex]\[ \left(2^{-4} \cdot z^{-3}\right)^5 \][/tex]
### Step 1: Apply the Power of a Product Rule
The rule for raising a product to a power states:
[tex]\[ (a \cdot b)^m = a^m \cdot b^m \][/tex]
We apply this rule to each term inside the parentheses.
Thus:
[tex]\[ \left(2^{-4} \cdot z^{-3}\right)^5 = (2^{-4})^5 \cdot (z^{-3})^5 \][/tex]
### Step 2: Simplify the Exponents
Next, we multiply the exponents inside the parentheses by the outer exponent:
For [tex]\( 2^{-4} \)[/tex]:
[tex]\[ (2^{-4})^5 = 2^{-4 \cdot 5} = 2^{-20} \][/tex]
For [tex]\( z^{-3} \)[/tex]:
[tex]\[ (z^{-3})^5 = z^{-3 \cdot 5} = z^{-15} \][/tex]
Thus, the expression becomes:
[tex]\[ 2^{-20} \cdot z^{-15} \][/tex]
### Step 3: Convert Negative Exponents to Positive Exponents
Recall that a negative exponent indicates the reciprocal:
[tex]\[ a^{-m} = \frac{1}{a^m} \][/tex]
We apply this property to each term in the expression:
For [tex]\( 2^{-20} \)[/tex]:
[tex]\[ 2^{-20} = \frac{1}{2^{20}} \][/tex]
For [tex]\( z^{-15} \)[/tex]:
[tex]\[ z^{-15} = \frac{1}{z^{15}} \][/tex]
Thus:
[tex]\[ 2^{-20} \cdot z^{-15} = \frac{1}{2^{20}} \cdot \frac{1}{z^{15}} \][/tex]
Which simplifies to:
[tex]\[ \frac{1}{2^{20} \cdot z^{15}} \][/tex]
### Conclusion
Therefore, the equivalent expression is:
[tex]\[ \boxed{\frac{1}{2^{20} \cdot z^{15}}} \][/tex]
This corresponds to the answer:
(A) [tex]\(\frac{1}{2^{20} \cdot z^{15}}\)[/tex]
