To simplify the given expression:

$\left( \frac{4mn}{m^{-2}n^6} \right)^{-2}$

we follow these steps:

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Step 1: Simplify the base of the fraction

The numerator is $4mn$, and the denominator is $m^{-2}n^6$. Simplify by applying the rules of exponents:

$\frac{4mn}{m^{-2}n^6} = 4m^{1 - (-2)}n^{1 - 6} = 4m^3n^{-5}.$

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Step 2: Apply the negative exponent to the entire fraction

The negative exponent $(-2)$ means we take the reciprocal and square the fraction:

$\left( 4m^3n^{-5} \right)^{-2} = \frac{1}{(4m^3n^{-5})^2}.$

Simplify the square:

$(4m^3n^{-5})^2 = 4^2m^{3 \cdot 2}n^{-5 \cdot 2} = 16m^6n^{-10}.$

Thus, taking the reciprocal:

$\frac{1}{16m^6n^{-10}} = \frac{n^{10}}{16m^6}.$

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Final Answer:

The equivalent expression is:

$\boxed{\frac{n^{10}}{16m^6}}$

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Verification:

1. Start by confirming the simplifications of the exponents were done correctly.
2. Ensure the exponent $-2$ was applied to both numerator and denominator correctly.

Would you like further clarification on any of these steps?

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Relative Questions:

1. How do the laws of exponents apply to negative bases?
2. Can this approach be extended to fractional exponents?
3. What is the impact of swapping numerator and denominator when using a negative exponent?
4. What happens if $n = 0$ or $m = 0$ despite the assumption $m \neq 0, n \neq 0$?
5. How does the negative exponent affect more complex expressions involving roots?

Tip:

Always simplify the base expression fully before applying an exponent to avoid errors in calculations.