To simplify the expression \((2ry^3)^4 \times (r^2y)^3\), we will apply the properties of integer exponents step by step.
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Applying the Power of a Product Rule: This rule states that \((ab)^n = a^n b^n\). \[ (2ry^3)^4 = (2^4)(r^4)(y^{3 \cdot 4}) = 16r^4y^{12} \]
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Now for the second part: \[ (r^2y)^3 = (r^2)^3(y)^3 = r^{2 \cdot 3}(y^3) = r^6y^3 \]
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Now we will multiply the two results: \[ 16r^4y^{12} \times r^6y^3 \]
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Combine the coefficients, and apply the product of powers rule for \(r\) and \(y\): \[ 16 \times (r^4 \times r^6) \times (y^{12} \times y^3) = 16r^{4+6}y^{12+3} = 16r^{10}y^{15} \]
Thus, the simplified expression is: \[ \boxed{16r^{10}y^{15}} \]