Let's simplify the expression step by step.

Given:

$\left(-\frac{15x^{-5}y^9}{-5z^{-4}}\right)^{-3}$

Step 1: Simplify the expression inside the parentheses.

1. Simplify the coefficients:

$\frac{-15}{-5} = 3$

So, the expression becomes:

$\frac{3x^{-5}y^9}{z^{-4}}$

2. Simplify the powers of $z$ by moving the negative exponent to the numerator:

$\frac{3x^{-5}y^9z^{4}}{1}$

This simplifies to:

$3x^{-5}y^9z^{4}$

Step 2: Apply the exponent $-3$ to the entire expression.

When raising an expression to a negative power, we flip the expression and change the sign of the exponent:

$\left(3x^{-5}y^9z^{4}\right)^{-3} = \frac{1}{\left(3x^{-5}y^9z^{4}\right)^{3}}$

Step 3: Expand the exponent over the product:

1. Cube the coefficient:

$3^3 = 27$

2. Multiply the exponents for each variable:

$\left(x^{-5}\right)^3 = x^{-15}$ $\left(y^{9}\right)^3 = y^{27}$ $\left(z^{4}\right)^3 = z^{12}$

So, the expression becomes:

$\frac{1}{27x^{-15}y^{27}z^{12}}$

Step 4: Simplify the expression by moving negative exponents to the numerator:

$\frac{x^{15}}{27y^{27}z^{12}}$

Therefore, the final simplified expression is:

$\frac{x^{15}}{27y^{27}z^{12}}$

Would you like more details on any part of this solution, or do you have any questions?

Here are 5 related questions for further exploration:

1. How would you simplify the expression if the exponent was positive instead of negative?
2. What are the general rules for multiplying and dividing exponents with different bases?
3. How do you simplify expressions with negative exponents in both the numerator and denominator?
4. Can you solve similar expressions with fractional exponents?
5. How does the simplification process change if you have more variables involved?

Tip: When dealing with complex exponents, break the problem down step by step, simplifying coefficients first, followed by variables, and then apply the exponents.