To find the greatest common factor (GCF) of the expressions \(12a^4b^7\) and \(18a^5b^3\), we will first determine the GCF of the coefficients (the numerical parts) and then the GCF of the variable parts.
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Finding the GCF of the coefficients (12 and 18):
- The prime factorization of 12 is \(2^2 \times 3\).
- The prime factorization of 18 is \(2 \times 3^2\).
- The GCF is found by taking the lowest power of each prime factor that appears in both factorizations:
- For \(2\), the lowest power is \(2^1\).
- For \(3\), the lowest power is \(3^1\).
- Thus, the GCF of the coefficients is: \[ 2^1 \times 3^1 = 2 \times 3 = 6. \]
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Finding the GCF of the variable parts:
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For \(a\):
- The powers are \(4\) (from \(a^4\)) and \(5\) (from \(a^5\)).
- The GCF is \(a^{\min(4, 5)} = a^4\).
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For \(b\):
- The powers are \(7\) (from \(b^7\)) and \(3\) (from \(b^3\)).
- The GCF is \(b^{\min(7, 3)} = b^3\).
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Combining the results:
- The overall GCF is obtained by multiplying the GCF of the coefficients by the GCF of the variable parts.
- Therefore, we have: \[ \text{GCF} = 6a^4b^3. \]
Thus, the greatest common factor of \(12a^4b^7\) and \(18a^5b^3\) is \(\boxed{6a^4b^3}\).
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