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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!

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.