To simplify the expression \(\frac{15x^7z^5 - 25xz^5}{-3x^3z^4}\), follow these steps:

### Step 1: Distribute the denominator across each term in the numerator

\[
\frac{15x^7z^5}{-3x^3z^4} - \frac{25xz^5}{-3x^3z^4}
\]

### Step 2: Simplify each fraction separately

**Simplifying the first fraction:**

\[
\frac{15x^7z^5}{-3x^3z^4}
\]

- Simplify the coefficients: \(\frac{15}{-3} = -5\)
- Simplify the powers of \(x\): \(x^7 \div x^3 = x^{7-3} = x^4\)
- Simplify the powers of \(z\): \(z^5 \div z^4 = z^{5-4} = z\)

So, the first term simplifies to:

\[
-5x^4z
\]

**Simplifying the second fraction:**

\[
\frac{25xz^5}{-3x^3z^4}
\]

- Simplify the coefficients: \(\frac{25}{-3} = -\frac{25}{3}\)
- Simplify the powers of \(x\): \(x \div x^3 = x^{1-3} = x^{-2}\)
- Simplify the powers of \(z\): \(z^5 \div z^4 = z^{5-4} = z\)

So, the second term simplifies to:

\[
-\frac{25}{3}x^{-2}z = -\frac{25z}{3x^2}
\]

### Step 3: Combine the simplified terms

\[
\frac{15x^7z^5 - 25xz^5}{-3x^3z^4} = -5x^4z + \frac{25z}{3x^2}
\]

So, the final simplified expression is:

\[
\boxed{-5x^4z + \frac{25z}{3x^2}}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you simplify expressions with negative exponents?
2. What happens if you factor out the common term before simplifying the expression?
3. How would the simplification change if the powers of \(x\) or \(z\) were different?
4. Can you rewrite the simplified expression in a different form, such as using positive exponents only?
5. How does this simplification relate to polynomial division?

### Tip:
When simplifying algebraic fractions, always reduce the coefficients and exponents carefully, and watch out for negative signs to avoid errors.

Learn how to simplify the algebraic expression (15x^7z^5 - 25xz^5) / (-3x^3z^4) step-by-step. This problem involves simplifying fractions with variables, covering polynomial division and exponent rules. Suitable for high school students studying algebra.

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