To simplify the expression $\left( -\frac{x^{-6}}{x^{-2}} \right)^{-2}$, let's break it down step-by-step:

1. Simplify the expression inside the parentheses: $\frac{x^{-6}}{x^{-2}} = x^{-6 - (-2)} = x^{-6 + 2} = x^{-4}$ So the expression becomes: $\left( -x^{-4} \right)^{-2}$

2. Apply the negative exponent outside the parentheses: When raising a term to a negative power, we take the reciprocal and change the sign of the exponent. Here we have: $\left( -x^{-4} \right)^{-2} = \frac{1}{(-x^{-4})^2}$

3. Square the expression inside the denominator: When squaring $-x^{-4}$, we get: $(-x^{-4})^2 = x^{-8}$ So now we have: $\frac{1}{x^{-8}}$

4. Simplify the final expression: Since $\frac{1}{x^{-8}} = x^8$, the simplified form is: $x^8$

Final Answer:

$x^8$

Would you like a detailed explanation of any step, or do you have any further questions?

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Here are some related questions to explore:

1. How do you simplify expressions with negative exponents in general?
2. What are the rules for handling negative exponents in fraction form?
3. How does squaring a negative exponent affect the result?
4. How do you handle expressions with multiple layers of exponents?
5. What is the effect of an outer negative exponent on an entire expression?

Tip: Remember, when dealing with negative exponents, flipping the fraction (taking the reciprocal) is a powerful simplification tool.