To simplify \( (5x^2 y^4)^3 \), we apply the power of a product rule, which states that \( (ab)^n = a^n b^n \), and also the power of a power rule, which states that \( (a^m)^n = a^{mn} \).
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Apply the power of the product: \[ (5x^2 y^4)^3 = 5^3 (x^2)^3 (y^4)^3 \]
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Calculate each part: \[ 5^3 = 125 \] \[ (x^2)^3 = x^{2 \times 3} = x^6 \] \[ (y^4)^3 = y^{4 \times 3} = y^{12} \]
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Combine all the parts: \[ (5x^2 y^4)^3 = 125 x^6 y^{12} \]
So, the simplified form is: \[ \boxed{125 x^6 y^{12}} \]