To simplify the expression \((\frac{y^3 \cdot 2yx^2}{2xy^3})^4\), we'll follow these steps:
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Simplify the fraction inside the parentheses: \[ \frac{y^3 \cdot 2yx^2}{2xy^3} = \frac{2y^{3+1}x^2}{2xy^3} \] The \(2\) in the numerator and the \(2\) in the denominator cancel out: \[ = \frac{y^4x^2}{xy^3} \] Now, we can separate the \(y\) terms and the \(x\) terms: \[ = \frac{y^4}{y^3} \cdot \frac{x^2}{x} \] Using the rule \(a^m / a^n = a^{m-n}\): \[ = y^{4-3} \cdot x^{2-1} = y^1 \cdot x^1 = yx \]
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Raise the simplified expression to the power of 4: \[ (yx)^4 \] Using the power of a product rule \((ab)^n = a^n b^n\): \[ = y^4 x^4 \]
Thus, the simplified expression is: \[ \boxed{y^4 x^4} \]