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].
