Find the greatest common factor. \[ 2 a^{3}, 7 a^{2} \] Write your answer as a constant times a product of single variables rasised to exponents.

To find the greatest common factor (GCF) of the two terms \(2 a^{3}\) and \(7 a^{2}\), we analyze both the numerical coefficients and the variable parts. The coefficients are 2 and 7, and their GCF is 1, as they don't share any common factors. For the variable part, \(a^{3}\) and \(a^{2}\), the GCF is \(a^{2}\) since it is the lower exponent. Therefore, combining these we have: \[ GCF = 1 \cdot a^{2} = a^{2} \] Our greatest common factor is \(a^{2}\). And just like that, we’ve uncovered the GCF without any fuss! This GCF might be simple, but it can power up your polynomial equations like a true superhero—making calculations neat and tidy. When tackling similar problems, remember to break things down into smaller pieces: focus on coefficients first, then tackle the variables. It makes finding the GCF smoother than a well-mixed smoothie! Just watch out for those tricky coefficients that might seem related but aren't!