To factor the expression \( 81m + 48mn \) completely, we first look for the greatest common factor (GCF) of the terms.
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Identify the GCF of the coefficients:
- The coefficients are 81 and 48. The GCF of 81 and 48 is 3.
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Identify the common variable:
- Both terms have the variable \( m \) in common.
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Combine the GCF from the coefficients and the common variable:
- The GCF is \( 3m \).
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Factor \( 3m \) out of the expression: \[ 81m + 48mn = 3m(27 + 16n) \]
So, the expression \( 81m + 48mn \) factored completely is: \[ \boxed{3m(27 + 16n)} \]