Understanding 10,000*100,000=1,000,000,000 helps understand (x^a)(x^b)=x^{a+b}.
Knowing how to multiply (1537)(824) helps to understand how to multiply (x^3+5x^2+3x+7)(8x^2+2x+4). The former is just the latter with x=10, and in fact the latter is easier, which is a common occurrence in mathematics where abstraction makes things easier. You should instinctively understand distributivity.
Arithmetic can be a warmup to (pre)algebra, but a conceptual thinker may do well just to get stuck into the (pre)algebra to get the conceptual underpinnings, and then backfill the arithmetic.
