The expression you've provided is:

\[
(x - 6y^3)^{-9}z^4
\]

This expression consists of two terms:

1. \( (x - 6y^3)^{-9} \)
2. \( z^4 \)

### Breaking Down the Expression:
- The term \( (x - 6y^3)^{-9} \) represents the reciprocal of \( (x - 6y^3) \) raised to the 9th power. In other words:
  \[
  (x - 6y^3)^{-9} = \frac{1}{(x - 6y^3)^9}
  \]
  
- The term \( z^4 \) is simply \( z \) raised to the 4th power.

### Combined Expression:
So, the expression can be rewritten as:

\[
\frac{z^4}{(x - 6y^3)^9}
\]

This is the simplified form of the expression.

Would you like further details on how to manipulate such expressions, or do you have any other questions? Here are some related questions you might find interesting:

1. How do you simplify expressions with negative exponents?
2. What is the general rule for dealing with expressions like \( (a + b)^{-n} \)?
3. How would you expand \( (x - y)^9 \) using the binomial theorem?
4. How do exponents distribute over multiplication and division?
5. What are some common mistakes to avoid when simplifying expressions with multiple terms and exponents?

**Tip:** Always remember that a negative exponent indicates a reciprocal. This is useful for simplifying complex expressions!

Explore the expression (x - 6y^3)^(-9) z^4 in algebra, including its simplification process and key concepts like exponents, reciprocals, and the binomial theorem. Suitable for advanced high school students and anyone interested in algebraic manipulations.

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