To find the greatest common factor (GCF) of \(18a^4b^3\) and \(21a^8b^3\), we start by determining the GCF of the numerical coefficients and then the GCF of each variable part.
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Numerical Coefficients:
- The coefficients are 18 and 21.
- The prime factorization of 18 is: \[ 18 = 2 \times 3^2 \]
- The prime factorization of 21 is: \[ 21 = 3 \times 7 \]
- The common factor from the prime factorizations is \(3\).
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Variable Part:
- For \(a^4\) and \(a^8\), the GCF is the variable raised to the lowest exponent: \[ \text{GCF}(a^4, a^8) = a^{\min(4, 8)} = a^4 \]
- For \(b^3\) in both terms, since they are the same, the GCF is: \[ \text{GCF}(b^3, b^3) = b^3 \]
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Combine the GCFs:
- Combining all these, we get: \[ \text{GCF}(18a^4b^3, 21a^8b^3) = 3 \cdot a^4 \cdot b^3 \]
Thus, the greatest common factor of \(18a^4b^3\) and \(21a^8b^3\) is: \[ \boxed{3a^4b^3} \]
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