Identify the greatest common factor of 12a^4b^7 and 18a^5b^3

Answer 1

To find the greatest common factor (GCF) of the expressions \(12a^4b^7\) and \(18a^5b^3\), we can break it down step by step.

1. Find the GCF of the coefficients:

   • The coefficients are 12 and 18.
   • The factors of 12 are: \(1, 2, 3, 4, 6, 12\)
   • The factors of 18 are: \(1, 2, 3, 6, 9, 18\)
   • The common factors are \(1, 2, 3, 6\).
   • Therefore, the GCF of 12 and 18 is 6.

2. Find the GCF of the variables \(a^4\) and \(a^5\):

   • The GCF for \(a^n\) terms is \(a^{\min(m,n)}\).
   • Here, \(m = 4\) and \(n = 5\), so \(\min(4, 5) = 4\).
   • Thus, the GCF of \(a^4\) and \(a^5\) is \(a^4\).

3. Find the GCF of the variables \(b^7\) and \(b^3\):

   • Again, we use \(b^{\min(m,n)}\).
   • Here, \(m = 7\) and \(n = 3\), so \(\min(7, 3) = 3\).
   • Therefore, the GCF of \(b^7\) and \(b^3\) is \(b^3\).

4. Combine the GCFs:

   • Putting it all together, we have: \[ \text{GCF} = 6a^4b^3 \]

Thus, the greatest common factor of \(12a^4b^7\) and \(18a^5b^3\) is \(\boxed{6a^4b^3}\).