Select the equivalent expression.

[tex]\[
\left(5^4 \cdot b^{-10}\right)^{-6} = ?
\][/tex]

Choose one answer:
A. [tex]\(\frac{b^{60}}{5^{24}}\)[/tex]
B. [tex]\(5^4 \cdot b^{60}\)[/tex]
C. [tex]\(5^{24} \cdot b^{60}\)[/tex]



Answer :

Sure, let's solve the given problem step-by-step.

We need to find the equivalent expression of:
[tex]\[ \left(5^4 \cdot b^{-10}\right)^{-6} \][/tex]

Step-by-Step Solution:

1. Understand the given expression: We have [tex]\(\left(5^4 \cdot b^{-10}\right)^{-6}\)[/tex].

2. Apply the power rule: The power rule states that [tex]\((a^m \cdot b^n)^p = a^{mp} \cdot b^{np}\)[/tex]. We can break this down into:
[tex]\[ (5^4)^{-6} \cdot (b^{-10})^{-6} \][/tex]

3. Simplify each part separately:
- For [tex]\(5^4\)[/tex]:
[tex]\[ (5^4)^{-6} = 5^{4 \cdot (-6)} = 5^{-24} \][/tex]
- For [tex]\(b^{-10}\)[/tex]:
[tex]\[ (b^{-10})^{-6} = b^{-10 \cdot (-6)} = b^{60} \][/tex]

4. Combine the simplified expressions: Now we have:
[tex]\[ 5^{-24} \cdot b^{60} \][/tex]

5. Express it in fraction form: Since [tex]\(5^{-24}\)[/tex] is the same as [tex]\(\frac{1}{5^{24}}\)[/tex], we get:
[tex]\[ \frac{b^{60}}{5^{24}} \][/tex]

So, the equivalent expression is:
[tex]\[ \boxed{\frac{b^{60}}{5^{24}}} \][/tex]

6. Choose the correct answer: Comparing this with the answer choices given:

(A) [tex]\(\frac{b^{60}}{5^{24}}\)[/tex] \
(B) [tex]\(5^4 \cdot b^{60}\)[/tex] \
(C) [tex]\(5^{24} \cdot b^{60}\)[/tex]

The correct answer is (A) [tex]\(\frac{b^{60}}{5^{24}}\)[/tex].

Therefore, the correct choice is [tex]\(\boxed{A}\)[/tex].