To simplify the expression \((3mn^2)^2(2m^0n)^3(3mn^2)^2(2m^0n)^3\), we will first simplify each part step by step:
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First term: \((3mn^2)^2\): \[ (3mn^2)^2 = 3^2 \cdot m^2 \cdot (n^2)^2 = 9m^2n^4 \]
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Second term: \((2m^0n)^3\): Since \(m^0 = 1\): \[ (2m^0n)^3 = (2 \cdot 1 \cdot n)^3 = 2^3 \cdot n^3 = 8n^3 \]
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Third term: \((3mn^2)^2\) (same as first term): \[ (3mn^2)^2 = 9m^2n^4 \]
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Fourth term: \((2m^0n)^3\) (same as second term): \[ (2m^0n)^3 = 8n^3 \]
Now we can combine all the simplified terms together: \[ (9m^2n^4) \cdot (8n^3) \cdot (9m^2n^4) \cdot (8n^3) \]
First, calculate the coefficients: \[ 9 \cdot 8 \cdot 9 \cdot 8 = 576 \]
Next, combine the variables:
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For \(m\): \[ m^2 \cdot m^2 = m^{2+2} = m^4 \]
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For \(n\): \[ n^4 \cdot n^3 \cdot n^4 \cdot n^3 = n^{4+3+4+3} = n^{14} \]
Thus, we have: \[ 576m^4n^{14} \]
So, the fully simplified expression is: \[ \boxed{576m^4n^{14}} \]