To simplify the expression $\frac{((a^{-2})^{-1})^{-1}}{\left(\frac{a}{a^{-1}}\right)^2}$, let’s work through it step-by-step.

Step 1: Simplify the Numerator $((a^{-2})^{-1})^{-1}$

1. Start with $(a^{-2})^{-1}$:

   • When raising a power to another power, multiply the exponents.
   • So, $(a^{-2})^{-1} = a^{2}$.

2. Now, take $(a^{2})^{-1}$:

   • This becomes $a^{-2}$.

Thus, the numerator simplifies to $a^{-2}$.

Step 2: Simplify the Denominator $\left(\frac{a}{a^{-1}}\right)^2$

1. Simplify the inner expression $\frac{a}{a^{-1}}$:

   • Dividing by $a^{-1}$ is equivalent to multiplying by $a$, so $\frac{a}{a^{-1}} = a \cdot a = a^2$.

2. Now, raise $a^2$ to the power of $2$:

   • $(a^2)^2 = a^{4}$.

So, the denominator simplifies to $a^{4}$.

Step 3: Combine the Results

Now, the expression becomes: $\frac{a^{-2}}{a^{4}}$

Using the rule $a^m / a^n = a^{m - n}$, we get: $a^{-2 - 4} = a^{-6}$

Final Answer

$a^{-6} = \frac{1}{a^6}$

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Would you like further details, or do you have any questions about these steps?

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Here are some related questions to explore:

1. What are the rules for handling negative exponents in different expressions?
2. How would the answer change if the base $a$ was a fraction?
3. Can we extend these simplification techniques to other bases or variables?
4. What happens if an exponent is zero? How does that affect an expression?
5. How does simplifying complex expressions differ when variables have unknown values?

Tip: Always simplify each part of an expression separately before combining them for a final answer.