Select the equivalent expression.

[tex]\[ \left(2^{-4} \cdot z^{-3}\right)^5 = ? \][/tex]

Choose one answer:
A. [tex]\( 2 z^2 \)[/tex]
B. [tex]\( \frac{1}{2^{20} \cdot z^{15}} \)[/tex]
C. [tex]\( 2^{20} \cdot z^{15} \)[/tex]



Answer :

To solve the given expression [tex]\(\left(2^{-4} \cdot z^{-3}\right)^5\)[/tex], we need to apply the properties of exponents. Here's a detailed step-by-step solution:

1. Understand the properties of exponents:

- Power of a product: [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex].
- Power of a power: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].

2. Apply the power of a product property:

[tex]\[ \left(2^{-4} \cdot z^{-3}\right)^5 = \left(2^{-4}\right)^5 \cdot \left(z^{-3}\right)^5 \][/tex]

3. Apply the power of a power property:

[tex]\[ \left(2^{-4}\right)^5 = 2^{-4 \cdot 5} = 2^{-20} \][/tex]
[tex]\[ \left(z^{-3}\right)^5 = z^{-3 \cdot 5} = z^{-15} \][/tex]

4. Combine the results:

[tex]\[ \left(2^{-4} \cdot z^{-3}\right)^5 = 2^{-20} \cdot z^{-15} \][/tex]

5. Simplify the expression:

Recall that [tex]\(a^{-m} = \frac{1}{a^m}\)[/tex]. Using this property:

[tex]\[ 2^{-20} \cdot z^{-15} = \frac{1}{2^{20}} \cdot \frac{1}{z^{15}} = \frac{1}{2^{20} \cdot z^{15}} \][/tex]

Hence, the equivalent expression is:

[tex]\[ \frac{1}{2^{20} \cdot z^{15}} \][/tex]

So, the correct answer is:
(B) [tex]\(\frac{1}{2^{20} \cdot z^{15}}\)[/tex].