https://file.aiquickdraw.com/mathbot/file/1728250346140sr29e0fs.jpg

The expression provided in the image is:

\[
\frac{35x^5 z^2 - 28x^5 z^6}{-4x^4 z^3}
\]

Let's solve it step by step.

### Step 1: Break the expression into two terms
We can split the fraction into two separate terms:

\[
\frac{35x^5 z^2}{-4x^4 z^3} - \frac{28x^5 z^6}{-4x^4 z^3}
\]

### Step 2: Simplify each term
Now, we simplify each term individually.

#### First term: 
\[
\frac{35x^5 z^2}{-4x^4 z^3}
\]

- Simplify the coefficients: \( \frac{35}{-4} = -\frac{35}{4} \)
- Simplify the \( x \) terms: \( \frac{x^5}{x^4} = x^{5-4} = x^1 = x \)
- Simplify the \( z \) terms: \( \frac{z^2}{z^3} = z^{2-3} = z^{-1} = \frac{1}{z} \)

So the first term simplifies to:
\[
-\frac{35}{4} \cdot \frac{x}{z} = -\frac{35x}{4z}
\]

#### Second term: 
\[
\frac{28x^5 z^6}{-4x^4 z^3}
\]

- Simplify the coefficients: \( \frac{28}{-4} = -7 \)
- Simplify the \( x \) terms: \( \frac{x^5}{x^4} = x^{5-4} = x^1 = x \)
- Simplify the \( z \) terms: \( \frac{z^6}{z^3} = z^{6-3} = z^3 \)

So the second term simplifies to:
\[
-7x z^3
\]

### Step 3: Combine the two terms
Now, combine the two simplified terms:

\[
-\frac{35x}{4z} + 7x z^3
\]

This is the simplified form of the given expression.

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Would you like to go over any specific part of this explanation, or have any further questions?

### Related Questions:
1. How do you handle exponents when dividing variables with the same base?
2. Can this expression be factored further?
3. What are some real-life applications of rational expressions like this one?
4. How would the result change if we were multiplying instead of dividing?
5. How does the negative sign affect each term during the simplification process?

### Tip:
Always simplify expressions by breaking them into smaller terms and working through one operation at a time, especially when dealing with variables and exponents.

Divide: (35x^5 z^2 - 28x^5 z^6) / (-4x^4 z^3)

Learn how to simplify the rational expression (35x^5 z^2 - 28x^5 z^6) divided by (-4x^4 z^3) using algebraic techniques. This problem covers key concepts like exponents, polynomial division, and rational expressions, ideal for students in Grades 9-11.

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