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The given expression to simplify is:

\[
\frac{4(4x - 5)^{1/5} - (x - 4)(4x - 5)^{-4/5}}{(4x - 5)^{4/5}}.
\]

### Step 1: Factor out common terms in the numerator
To simplify, first factor out the common term \((4x - 5)^{-4/5}\) from the numerator.

\[
4(4x - 5)^{1/5} = 4(4x - 5)^{5/5 \times 1/5} = 4(4x - 5)^{-4/5} \times (4x - 5).
\]

\[
(x - 4)(4x - 5)^{-4/5} = (x - 4)(4x - 5)^{-4/5}.
\]

Now, the numerator can be rewritten as:

\[
(4(4x - 5) - (x - 4))(4x - 5)^{-4/5} 
\]

### Step 2: Simplify the fraction
The expression now becomes:

\[
\frac{4(4x - 5) - (x - 4)}{(4x - 5)^{4/5}}.
\]

Since the common factor \((4x - 5)^{4/5}\) divides out with \((4x - 5)\) resulting in:

### Step 3: Simplify numerator
The numerator simplifies to: 

\[
\frac{4(4x - 5)(-x-4)}{(4x-5)^1} = x+4 
\] 

We simplify to expression, the result turns into \[(x-4).


Simplify the rational expression: 4(4x - 5)^{1/5} - (x - 4)(4x - 5)^{-4/5} / (4x - 5)^{4/5}.

Learn how to simplify the rational expression 4(4x - 5)^{1/5} - (x - 4)(4x - 5)^{-4/5} / (4x - 5)^{4/5}. This problem involves key concepts of factoring and the laws of exponents, suitable for high school students in grades 10-12.

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