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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:
\[
\frac{a c^7}{4 b^3}
\]

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Here are five related questions to expand your understanding:
1. How would the

Simplify the following expression involving negative exponents: \frac{4^{-1}a^{-4}b^2c^5}{a^{-5}b^5c^{-2}}

Learn how to simplify the algebraic expression involving negative exponents: 4^{-1}a^{-4}b^2c^5 / a^{-5}b^5c^{-2}. This step-by-step guide will walk you through using the properties of exponents and simplifying complex expressions. Suitable for Grades 8-10.

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