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.

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 $(xyz)^n$.