The problem asks us to simplify the expression $(5^4 \cdot 6^{-10})^{-6}$. We must choose the equivalent expression from the options given.

1. Problem Description

The problem asks us to simplify the expression $(5^4 \cdot 6^{-10})^{-6}$. We must choose the equivalent expression from the options given.

2. Solution Steps

We need to simplify the expression $(5^4 \cdot 6^{-10})^{-6}$. We can use the power of a product rule, which states that $(ab)^n = a^n b^n$.

$(5^4 \cdot 6^{-10})^{-6} = (5^4)^{-6} \cdot (6^{-10})^{-6}$

We can use the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$.

$(5^4)^{-6} = 5^{4 \cdot (-6)} = 5^{-24}$

$(6^{-10})^{-6} = 6^{-10 \cdot (-6)} = 6^{60}$

So the expression becomes $5^{-24} \cdot 6^{60}$.

Since $a^{-n} = \frac{1}{a^n}$, we can rewrite $5^{-24}$ as $\frac{1}{5^{24}}$.

Therefore, $5^{-24} \cdot 6^{60} = \frac{1}{5^{24}} \cdot 6^{60} = \frac{6^{60}}{5^{24}}$.

3. Final Answer

The equivalent expression is $\frac{6^{60}}{5^{24}}$. The correct answer is (C).