To factor out the greatest common factor (GCF) from the expression \( 24ab^2c + 18a^3b \), we first need to determine the GCF of the coefficients and the variables.
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Find the GCF of the coefficients (24 and 18):
- The prime factorization of 24 is \( 2^3 \times 3 \).
- The prime factorization of 18 is \( 2 \times 3^2 \).
- The GCF of 24 and 18 is obtained by taking the lowest power of each prime factor present in both factorizations:
- For \( 2 \), the minimum power is \( 2^1 = 2 \).
- For \( 3 \), the minimum power is \( 3^1 = 3 \).
- Thus, the GCF of 24 and 18 is \( 2^1 \times 3^1 = 6 \).
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Find the GCF of the variables:
- For \( ab^2c \) and \( a^3b \):
- The lowest power of \( a \) is \( a^1 \) (since \( b^2c \) has \( a^1 \) and \( a^3b \) has \( a^3 \)).
- The lowest power of \( b \) is \( b^1 \) (the first term has \( b^2 \) and the second has \( b^1 \)).
- The variable \( c \) appears only in the first term, so we do not include it in the GCF.
- Therefore, the GCF of the variables is \( ab \).
- For \( ab^2c \) and \( a^3b \):
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Combine the GCF of coefficients and the variables:
- The overall GCF of \( 24ab^2c + 18a^3b \) is \( 6ab \).
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Factor out the GCF from the expression:
- Now we can rewrite the expression: \[ 24ab^2c + 18a^3b = 6ab(4bc + 3a^2) \]
Thus, the expression \( 24ab^2c + 18a^3b \) can be factored as: \[ 6ab(4bc + 3a^2) \]
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