Final answer:
To regroup and rename 3,200, one can represent it as 3 x 10^3 + 2 x 10^2, or alternatively as 32 x 10^2. This process relies on understanding the base-ten numbering system, where each place is ten times greater than the one to its right.
Step-by-step explanation:
To regroup and rename the number 3,200 in terms of base-ten blocks, one could break down the number into its powers of ten. The number 3,200 can be deconstructed to show that it consists of 3 thousands (3 x 103), 2 hundreds (2 x 102), and no tens or ones (0 x 101 and 0 x 100).
Regrouping and renaming this number means representing it in a different way while still retaining its original value. This could involve, for example, changing 3,200 into 32 hundreds (32 x 102), to show it purely as hundreds without thousands. Understanding that each place value in the numbering system is ten times greater than the one to its right helps in the regrouping process.
Remember that regrouping is not just about 'counting zeros' as each power of ten has an explicit value. So, while you could add two zeros to 32 to indicate hundreds, it's not the same as just starting with 10 and adding zeros. This is a fundamental principle of our base-ten counting system that stemmed from the fact humans typically have ten fingers, making it a natural numerical foundation.