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fraccion racional resultante en su forma mas simple

Let's simplify the given expression step by step. The expression is:

\[
\left( \frac{a^2 - 16a + 64}{a^2 - 64} \times \frac{a^3 - 9a^2 + 8a}{2a^2 - 128} \right) \div \frac{a^2 + a}{2}
\]

### Step 1: Factor each polynomial where possible.

1. **Numerator of the first fraction:** \(a^2 - 16a + 64\)
   - This can be factored as a perfect square trinomial: 
     \[
     a^2 - 16a + 64 = (a - 8)^2
     \]

2. **Denominator of the first fraction:** \(a^2 - 64\)
   - This is a difference of squares:
     \[
     a^2 - 64 = (a - 8)(a + 8)
     \]

3. **Numerator of the second fraction:** \(a^3 - 9a^2 + 8a\)
   - Factor out the common factor \(a\):
     \[
     a^3 - 9a^2 + 8a = a(a^2 - 9a + 8)
     \]
   - Factor \(a^2 - 9a + 8\):
     \[
     a^2 - 9a + 8 = (a - 8)(a - 1)
     \]
   - So, the numerator becomes:
     \[
     a(a - 8)(a - 1)
     \]

4. **Denominator of the second fraction:** \(2a^2 - 128\)
   - Factor out the common factor \(2\):
     \[
     2a^2 - 128 = 2(a^2 - 64) = 2(a - 8)(a + 8)
     \]

5. **Denominator of the third fraction:** \(a^2 + a\)
   - Factor out the common factor \(a\):
     \[
     a^2 + a = a(a + 1)
     \]

### Step 2: Substitute the factors back into the expression.

\[
\left( \frac{(a - 8)^2}{(a - 8)(a + 8)} \times \frac{a(a - 8)(a - 1)}{2(a - 8)(a + 8)} \right) \div \frac{a(a + 1)}{2}
\]

### Step 3: Simplify the expression.

1. Cancel out common factors:
   - In the first fraction: \(a - 8\) (cancel one from the numerator and denominator)
   - In the second fraction: \(a - 8\) (cancel one from the numerator and denominator)
   - \(a - 8\) is common in both fractions.

\[
\frac{(a - 8)}{(a + 8)} \times \frac{a(a - 1)}{2(a + 8)} \div \frac{a(a + 1)}{2}
\]

2. Multiply the fractions:

\[
\frac{(a - 8) \times a(a - 1)}{2(a + 8) \times (a + 8)} = \frac{a(a - 8)(a - 1)}{2(a + 8)^2}
\]

3. Division by \(\frac{a(a + 1)}{2}\) is equivalent to multiplying by the reciprocal:

\[
\frac{a(a - 8)(a - 1)}{2(a + 8)^2} \times \frac{2}{a(a + 1)} = \frac{2a(a - 8)(a - 1)}{2a(a + 1)(a + 8)^2}
\]

4. Cancel out common factors:
   - \(a\) and 2 are common factors in the numerator and denominator.

\[
\frac{(a - 8)(a - 1)}{(a + 1)(a + 8)^2}
\]

### Final Answer:

The simplified form of the given rational expression is:

\[
\frac{(a - 8)(a - 1)}{(a + 1)(a + 8)^2}
\]

Do you want further details or have any questions?

Here are five related questions to deepen your understanding:
1. How do you factor a perfect square trinomial?
2. What is the process for factoring a difference of squares?
3. How can you simplify complex rational expressions?
4. What are the common mistakes to avoid when factoring polynomials?
5. How do you determine the domain of a rational expression?

**Tip:** When simplifying rational expressions, always factor completely first to easily identify and cancel common factors.

Learn how to simplify the given rational expression step-by-step. Understand the process of factoring polynomials and cancelling common factors. Suitable for advanced high school students.

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