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The given expression to simplify is:

\[
\frac{(8y^7z^2 - 19y^7z^7)}{-2y^3z^4}
\]

### Step 1: Factor the numerator.
The numerator is \( 8y^7z^2 - 19y^7z^7 \). We can factor out the common terms \( y^7 \):

\[
y^7(8z^2 - 19z^7)
\]

So, the expression becomes:

\[
\frac{y^7(8z^2 - 19z^7)}{-2y^3z^4}
\]

### Step 2: Simplify the powers of \( y \).
Now, simplify the powers of \( y \) by dividing \( y^7 \) by \( y^3 \). Using the law of exponents, \( y^7 \div y^3 = y^{7-3} = y^4 \):

\[
\frac{y^4(8z^2 - 19z^7)}{-2z^4}
\]

### Step 3: Distribute the division across the terms inside the parentheses.
We now divide each term of \( 8z^2 - 19z^7 \) by \( -2z^4 \):

\[
\frac{8z^2}{-2z^4} - \frac{19z^7}{-2z^4}
\]

### Step 4: Simplify the terms.
1. \( \frac{8z^2}{-2z^4} = -4z^{-2} = -\frac{4}{z^2} \)
2. \( \frac{19z^7}{-2z^4} = -\frac{19}{2}z^3 \)

### Step 5: Combine everything.
The simplified expression is:

\[
y^4 \left( -\frac{4}{z^2} - \frac{19}{2}z^3 \right)
\]

Thus, the final simplified answer is:

\[
y^4 \left( -\frac{4}{z^2} - \frac{19}{2}z^3 \right)
\]

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Would you like any additional details or clarification on any of the steps? 

Here are some related questions:
1. How do you handle division of polynomials in multiple variables?
2. What are the rules for simplifying exponents during division?
3. How can factoring help in simplifying polynomial expressions?
4. What is the significance of negative exponents in a fraction?
5. How does distributing division across terms work in polynomial division?

**Tip:** Always remember to factor out common terms in both the numerator and denominator to simplify complex expressions more easily!

Divide (8y^7z^2 - 19y^7z^7) ÷ (-2y^3z^4). Simplify your answer as much as possible.

Learn how to simplify the algebraic expression (8y^7z^2 - 19y^7z^7) ÷ (-2y^3z^4) using factoring and exponent rules. Follow a step-by-step solution with a detailed breakdown of exponent manipulation and polynomial division. Suitable for high school students (Grades 9-12).

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