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The given expression is:

\[
\frac{7y^{-2}x^{-1}}{5x^{-2} + y^{-1}}
\]

We are tasked with simplifying it and rewriting the expression using only positive exponents.

Let’s break down the steps:

### Step 1: Simplify the numerator and denominator

- **Numerator**: \( 7y^{-2}x^{-1} \)
    - The term \( y^{-2} \) becomes \( \frac{1}{y^2} \).
    - The term \( x^{-1} \) becomes \( \frac{1}{x} \).

Thus, the numerator becomes:

\[
\frac{7}{y^2 x}
\]

- **Denominator**: \( 5x^{-2} + y^{-1} \)
    - The term \( x^{-2} \) becomes \( \frac{1}{x^2} \).
    - The term \( y^{-1} \) becomes \( \frac{1}{y} \).

Thus, the denominator becomes:

\[
\frac{5}{x^2} + \frac{1}{y}
\]

### Step 2: Combine the result

The expression is now:

\[
\frac{\frac{7}{y^2 x}}{\frac{5}{x^2} + \frac{1}{y}}
\]

We can now simplify this fraction. First, find the common denominator in the denominator:

\[
\frac{5}{x^2} + \frac{1}{y} = \frac{5y + x^2}{x^2 y}
\]

Now the overall expression becomes:

\[
\frac{7}{y^2 x} \div \frac{5y + x^2}{x^2 y}
\]

### Step 3: Simplify further

This division of fractions becomes:

\[
\frac{7}{y^2 x} \times \frac{x^2 y}{5y + x^2}
\]

Now cancel terms:

- \( y \) in the denominator cancels with one \( y \) in the numerator, leaving \( y \) in the denominator.
- \( x \) cancels one \( x \) from \( x^2 \), leaving \( x \) in the numerator.

Thus, the simplified expression is:

\[
\frac{7x}{y(5y + x^2)}
\]

This is the final simplified form, with all exponents positive.

Let me know if you'd like further clarification or details on any step!

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Here are 5 related questions:
1. How do you handle negative exponents when simplifying expressions?
2. What is the process for simplifying a complex fraction with multiple variables?
3. How do you find a common denominator in an expression with variables?
4. How do you multiply fractions and simplify the resulting expression?
5. What strategies can be used to cancel out terms in a fraction?

**Tip:** Always factor terms wherever possible to simplify expressions quickly and avoid errors.

Simplify the expression: (7y^{-2}x^{-1}) / (5x^{-2} + y^{-1}) and write the answer using only positive exponents.

This page explains how to simplify a rational expression with negative exponents: (7y^{-2}x^{-1}) / (5x^{-2} + y^{-1}). Step-by-step solutions using exponent rules, algebraic fractions, and rational expressions. Suitable for high school algebra students (Grades 9-12).

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