To simplify the expression \(3x^3y^{-4} - 12x^7y^4\), we first need to rewrite the terms in a more manageable form. We will factor out the common factors from both terms.
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Identify the common factors:
- The coefficient: The greatest common factor of \(3\) and \(-12\) is \(3\).
- For \(x\): The lowest exponent is \(x^3\) (from the first term).
- For \(y\): The lowest exponent is \(y^{-4}\) (from the first term).
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Factor out \(3x^3y^{-4}\): \[ 3x^3y^{-4}\left(1 - 4x^{4}y^{8}\right) \]
This step involves rewriting \(y^{-4}\) as \(\frac{1}{y^{4}}\):
- Final expression: \[ 3x^3 \cdot \frac{1}{y^4} - 12x^7y^4 = 3x^3y^{-4}(1 - 4x^{4}y^{8}) \]
None of the provided responses are exactly equivalent to this simplified expression. The original question was about generating equivalent algebraic expressions, but if you were looking for one that might logically follow processing, \(4x^4y^8\) seems to be the closest in relation as a manipulated version of the expression but not directly equivalent. You may want to confirm or check that the numbers provided are correct or complete.