Let's simplify the expression you provided, step by step:

The expression is:

$\frac{4^{-1}a^{-4}b^2c^5}{a^{-5}b^5c^{-2}}$

Step 1: Simplify the constant term

For the constant $4^{-1}$, recall that a negative exponent means taking the reciprocal: $4^{-1} = \frac{1}{4}$

Step 2: Simplify the exponents for each variable

For the $a$ terms, use the property of exponents: $\frac{a^{-4}}{a^{-5}} = a^{-4 - (-5)} = a^{-4 + 5} = a^1 = a$

For the $b$ terms: $\frac{b^2}{b^5} = b^{2-5} = b^{-3}$

For the $c$ terms: $\frac{c^5}{c^{-2}} = c^{5 - (-2)} = c^{5 + 2} = c^7$

Step 3: Combine all parts

Now put all the simplified parts together: $\frac{1}{4} \times a \times b^{-3} \times c^7 = \frac{a c^7}{4 b^3}$

Final Answer:

The simplified expression is: $\frac{a c^7}{4 b^3}$

Let me know if you would like more details or if you have any questions!

Here are five related questions to expand your understanding:

1. How would the