Final answer:
The greatest common factor (GCF) for the expressions 34m^3n^6 and m^2n^6 is m^2n^6, because it is the highest power of each variable present in both expressions. 34 is not included in the GCF as it does not divide m^2n^6. Therefore, the correct answer is option A: m^2n^6.
Step-by-step explanation:
The greatest common factor (GCF) of two or more expressions is the largest expression that divides each of the given expressions without leaving a remainder. To find the GCF of 34m^3n^6 and m^2n^6, we need to look for the highest power of each variable that appears in every term.
Breaking down each term:
- 34m^3n^6 = 2 × 17 × m^3 × n^6
- m^2n^6 = m^2 × n^6
Since 34 is a constant and is not a factor of m^2n^6, it is excluded from the GCF. The variable m is raised to the third power in the first term and the second power in the second term, so m^2 is the highest power of m that divides into both expressions. Similarly, n^6 is the highest power of n that divides into both expressions. Therefore, the GCF is m^2n^6, which is option A.