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We are tasked with simplifying the following expression and writing the answer using only positive exponents:

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

### Step-by-step simplification:

#### 1. Simplify the first part: \(\left( -3u^3 v^2 w \right)^{-2}\)

Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we distribute the exponent \(-2\) to each factor:

\[
(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}
\]
\[
(u^3)^{-2} = u^{3 \cdot (-2)} = u^{-6}
\]
\[
(v^2)^{-2} = v^{2 \cdot (-2)} = v^{-4}
\]
\[
(w)^{-2} = w^{-2}
\]

Thus, the first part becomes:
\[
\frac{1}{9} u^{-6} v^{-4} w^{-2}
\]

#### 2. Combine with the second part: \(u w^2\)

Now, multiply this result by \(u w^2\):

\[
\frac{1}{9} u^{-6} v^{-4} w^{-2} \cdot u^1 w^2
\]

#### 3. Combine like terms:

For \(u\): 
\[
u^{-6} \cdot u^1 = u^{-5}
\]

For \(w\): 
\[
w^{-2} \cdot w^2 = w^0 = 1
\]

So, we are left with:
\[
\frac{1}{9} u^{-5} v^{-4}
\]

#### 4. Write using only positive exponents:

To express the answer with positive exponents, we move terms with negative exponents to the denominator:

\[
\frac{1}{9 u^5 v^4}
\]

### Final answer:
\[
\boxed{\frac{1}{9 u^5 v^4}}
\]

Let me know if you would like further clarification or additional details.

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Here are 5 related questions for practice:
1. Simplify \(\left( 4a^2b^{-3} \right)^{-1} \cdot a^{-2}b^2\).
2. Simplify \(\left( -2x^3y^2 \right)^{-3} \cdot x^{-1}y^{-4}\).
3. How would you simplify \(\left(5x^2y^{-1}\right)^{-2}\)?
4. Combine and simplify \(\left(2a^{-2}b\right)^3 \cdot \left(a b^{-2}\right)\).
5. Simplify \(\left( 7p^3q^4r \right)^{-2} \cdot \left( p^2 r^{-3} \right)\).

**Tip:** Always remember to apply the exponent to every term inside the parentheses when simplifying expressions like \((xyz)^n\).

Simplify the expression: (-3u^3v^2w)^(-2)(uw^2), and write the answer using only positive exponents.

Learn how to simplify the expression (-3u^3v^2w)^(-2)(uw^2) using exponent rules, breaking down each step clearly to convert negative exponents into positive exponents. Suitable for Grades 9-12 students, this problem focuses on exponent laws and algebraic expressions.

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