Which expression is equivalent to \( \frac{(3m^{-1}n^2)^{-4}}{(2m^{-2}n)^3} \)?

What is the simplified form of the given expression?

The expression can be simplified as follows: \[ \frac{(3m^{-1}n^2)^{-4}}{(2m^{-2}n)^3} \] \[ = \frac{1}{(3m^{-1}n^2)^4 \times (2m^{-2}n)^{-3}} \] \[ = \frac{1}{3^4 \times m^{-1 \times 4} \times n^{2 \times 4} \times 2^{-3} \times m^{-2 \times -3} \times n^{-3}} \] \[ = \frac{1}{81m^{-4}n^{8} \times 8m^6n^{-3}} \] \[ = \frac{1}{648m^2n^{5}} \] Therefore, the expression is equivalent to \( \frac{1}{648m^2n^{5}} \).

Explanation:

Simplifying the given expression:
To simplify the given expression, we first apply the negative exponent property, which states that \( x^{-a} = \frac{1}{x^a} \). Using this property, we can rewrite the expression as shown above. Next, we simplify the terms inside the parentheses by raising each term to the power outside the parentheses. This involves multiplying the exponents together. After simplifying the terms inside the parentheses, we combine the terms in the numerator and denominator using the properties of exponents. Finally, after combining like terms, we arrive at the simplified form of the given expression, which is \( \frac{1}{648m^2n^{5}} \).
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