To simplify the expression \((2x^4)^3\), you can apply the power of a product rule, which states that \((ab)^n = a^n b^n\). Here, \(a = 2\) and \(b = x^4\), and \(n = 3\).
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Apply the rule: \[ (2x^4)^3 = 2^3 \cdot (x^4)^3 \]
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Calculate \(2^3\): \[ 2^3 = 8 \]
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Now simplify \((x^4)^3\) using the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\): \[ (x^4)^3 = x^{4 \cdot 3} = x^{12} \]
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Combine the results: \[ (2x^4)^3 = 8 \cdot x^{12} \]
Thus, the expression \((2x^4)^3\) simplifies to:
\[ \boxed{8x^{12}} \]