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Let's simplify the expression step-by-step while keeping all exponents positive.

The given expression is:

\[
\frac{5u^{-1}}{v^{-2} - u^{-1}}
\]

### Step 1: Convert negative exponents to positive
- \( u^{-1} = \frac{1}{u} \)
- \( v^{-2} = \frac{1}{v^2} \)

Now, rewrite the expression:

\[
\frac{\frac{5}{u}}{\frac{1}{v^2} - \frac{1}{u}}
\]

### Step 2: Simplify the denominator

To simplify the denominator, we need to combine the fractions. Find a common denominator for the terms in the denominator:

\[
\frac{1}{v^2} - \frac{1}{u} = \frac{u - v^2}{uv^2}
\]

So the expression becomes:

\[
\frac{\frac{5}{u}}{\frac{u - v^2}{uv^2}}
\]

### Step 3: Simplify the overall fraction

We can simplify by multiplying the numerator by the reciprocal of the denominator:

\[
\frac{5}{u} \times \frac{uv^2}{u - v^2} = \frac{5v^2}{u - v^2}
\]

### Final Answer:
The simplified expression, with only positive exponents, is:

\[
\frac{5v^2}{u - v^2}
\]

Would you like any further details or clarifications on this solution?

### Related Questions:
1. How do you handle negative exponents in multiplication and division?
2. What are the rules for simplifying expressions with both variables and exponents?
3. How do you add or subtract fractions with different denominators?
4. How can we simplify complex rational expressions further if required?
5. How do positive exponents relate to the original base in expressions?

### Tip:
When working with negative exponents, remember that they indicate reciprocals of the base raised to the corresponding positive exponent.

Simplify the expression: (5u^(-1)) / (v^(-2) - u^(-1)) and write your answer using only positive exponents.

Learn how to simplify the expression (5u^(-1)) / (v^(-2) - u^(-1)) by converting negative exponents into positive ones. This detailed solution demonstrates step-by-step fraction simplification for high school algebra students. Suitable for Grades 9-12.

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