To simplify the expression \((4x^4y^{-4})^3(x^7y^3)^2\), we'll use the properties of exponents.
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Expand each part separately:
\[ (4x^4y^{-4})^3 = 4^3(x^4)^3(y^{-4})^3 = 64x^{12}y^{-12} \]
\[ (x^7y^3)^2 = (x^7)^2(y^3)^2 = x^{14}y^6 \]
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Multiply the two results together:
Now we combine \(64x^{12}y^{-12}\) and \(x^{14}y^6\):
\[ 64x^{12}y^{-12} \cdot x^{14}y^6 = 64(x^{12} \cdot x^{14})(y^{-12} \cdot y^6) \]
Applying the properties of exponents (adding the exponents):
\[ x^{12 + 14} = x^{26}, \quad y^{-12 + 6} = y^{-6} \]
So we get:
\[ 64x^{26}y^{-6} \]
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Rewrite with positive exponents:
To express \(y^{-6}\) as a positive exponent, we rewrite the expression:
\[ 64x^{26} \cdot \frac{1}{y^6} = \frac{64x^{26}}{y^6} \]
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Final expression:
The equivalent algebraic expression is:
\[ \frac{64x^{26}}{y^6} \]
None of the provided options match this answer directly. However, if we were to match it closer to potential options reflecting simplification, we'd state that the above process results in a form that simplifies based on the final computation of exponents.
The correct simplified expression based on the rules provided is:
\[ \frac{64x^{26}}{y^6} \] (not explicitly in the original options provided). Please check if there might be some misunderstanding in the options or output required.