The given expression is:

$\left( \frac{x^4 y^6 z^5}{x^{-4} y^{-6} z^{-5}} \right)^{-2}$

Step 1: Simplify the expression inside the parentheses.

Apply the laws of exponents. Specifically, when dividing powers of the same base, subtract the exponents:

$\frac{x^4}{x^{-4}} = x^{4 - (-4)} = x^{4 + 4} = x^8$ $\frac{y^6}{y^{-6}} = y^{6 - (-6)} = y^{6 + 6} = y^{12}$ $\frac{z^5}{z^{-5}} = z^{5 - (-5)} = z^{5 + 5} = z^{10}$

So, the expression simplifies to:

$(x^8 y^{12} z^{10})^{-2}$

Step 2: Apply the exponent of $-2$.

To apply this exponent to each factor, multiply the exponents by $-2$:

$x^{8 \times (-2)} = x^{-16}$ $y^{12 \times (-2)} = y^{-24}$ $z^{10 \times (-2)} = z^{-20}$

Thus, the simplified expression is:

$x^{-16} y^{-24} z^{-20}$

Step 3: Write in standard form.

Since all the exponents are negative, you can rewrite the expression with positive exponents in the denominator:

$\frac{1}{x^{16} y^{24} z^{20}}$

So, the fully simplified expression is:

$\boxed{\frac{1}{x^{16} y^{24} z^{20}}}$

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. What are the laws of exponents, and how are they applied in simplifying expressions?
2. How do negative exponents affect the position of a base in a fraction?
3. What happens when you raise an exponential expression to a power?
4. Can you generalize the steps for simplifying any rational expression with exponents?
5. How would the result change if the original exponent outside the parentheses were positive?

Tip: When working with exponents, always remember to carefully apply the rules for multiplication, division, and raising powers, as they follow strict patterns.