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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.

Simplify the exponential expression. Assume that the variables represent nonzero real numbers: (x^4 * y^6 * z^5) / (x^-4 * y^-6 * z^-5))^-2

This problem simplifies the exponential expression ((x^4 * y^6 * z^5) / (x^-4 * y^-6 * z^-5))^-2. Learn the rules of exponents and how to apply them step-by-step to solve this algebraic expression, suitable for high school students.

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