The given expression is:

$\left( \frac{3 \cdot w^4}{w^3 \cdot n^5} \right)^{-1}$

To simplify this, we will first break it down step by step.

Step 1: Simplifying the fraction inside the parentheses

The fraction inside the parentheses is:

$\frac{3 \cdot w^4}{w^3 \cdot n^5}$

We can simplify the powers of $w$ by using the rule $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{w^4}{w^3} = w^{4-3} = w^1 = w$

So, the expression becomes:

$\frac{3 \cdot w}{n^5}$

Step 2: Applying the exponent of -1

Now, we have:

$\left( \frac{3 \cdot w}{n^5} \right)^{-1}$

To simplify this, apply the rule $\left( \frac{a}{b} \right)^{-1} = \frac{b}{a}$, so we get:

$\frac{n^5}{3 \cdot w}$

Final Answer:

$\frac{n^5}{3w}$

Would you like more details on any specific step or have other questions related to this? Here are some related questions to explore further:

1. How do you handle negative exponents in other expressions?
2. What is the rule for simplifying powers of the same variable with different exponents?
3. How do you deal with higher powers in fractions?
4. What would the expression look like if we had additional variables?
5. How can you simplify expressions with both positive and negative exponents?

Tip: Always simplify exponents first before applying any negative exponent rule to avoid confusion!