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The given expression to simplify is:

\[
\left( \frac{u^2}{w} \cdot \frac{u^{-1}v^3}{3w^{-3}} \right)^2
\]

We need to simplify this expression step by step and write the result with only positive exponents.

### Step 1: Combine the two fractions inside the parentheses.
Multiply the numerators and denominators:

\[
\frac{u^2}{w} \cdot \frac{u^{-1}v^3}{3w^{-3}} = \frac{u^2 \cdot u^{-1} \cdot v^3}{w \cdot 3 \cdot w^{-3}} = \frac{u^{2 + (-1)} \cdot v^3}{3 \cdot w^{1 + (-3)}}
\]

This simplifies to:

\[
\frac{u^{1} \cdot v^3}{3 \cdot w^4} = \frac{uv^3}{3w^4}
\]

### Step 2: Apply the exponent of 2 to the entire fraction.

Now, square both the numerator and the denominator:

\[
\left( \frac{uv^3}{3w^4} \right)^2 = \frac{(uv^3)^2}{(3w^4)^2}
\]

Simplifying both the numerator and denominator:

- Numerator: \((uv^3)^2 = u^2v^6\)
- Denominator: \((3w^4)^2 = 9w^8\)

So, the expression becomes:

\[
\frac{u^2v^6}{9w^8}
\]

This is the simplified result with only positive exponents.

### Final Answer:
\[
\frac{u^2v^6}{9w^8}
\]

Would you like more details on any step?

Here are 5 follow-up questions for practice:
1. What happens if the exponents were negative in the final answer?
2. How do you multiply terms with the same base but different exponents?
3. Why do we add exponents when multiplying powers with the same base?
4. What would the result be if we cubed the entire fraction instead of squaring it?
5. How would the answer change if \( w^{-3} \) was \( w^3 \)?

**Tip:** When simplifying exponents, always combine like terms first before applying additional exponents.

Simplify the given expression: \left( \frac{u^2}{w} \cdot \frac{u^{-1}v^3}{3w^{-3}} \right)^2. Write your answer using only positive exponents.

Learn how to simplify the algebraic expression \left( \frac{u^2}{w} \cdot \frac{u^{-1}v^3}{3w^{-3}} \right)^2. This involves using exponent rules to express the result with only positive exponents, making it suitable for students in Grades 9-11.

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