exponentiation - Why are negative exponents dividing instead of multiplying?

In trying to relearn scientific notation after years since I left school, I noticed that when we get a very small number and convert it to scientific notation, you're actually multiplying the tiny number a certain number of times so that the number becomes something between 1 and 10, and you write the inverse equation as scientific notation.
Example:

0.0035

I multiply it by 10³(or 1000) and then I have it written in scientific notation as 3.5 X 10^-3, which is not actually the formula describing the process to get scientific notation, but rather the formula that reverts it.

Now, when converting a large number- e.g 13,000,000 - into scientific notation, I figured out you have to divide it so as to get a number between 1 and 10, something that can't be achieved - I thought - if the exponentiation is doing successive multiplications.

So I ended up figuring out a negative exponent, to be capable of reverting the scientific notation described on the first example above, which turns a 3.5 back into 0.0035, had to be performing successive divisions.

So, for my second example I have to do 13,000,000 ÷ 10⁷ or 13,000,000 x ^-7, in either case getting 1.3, so that 13,000,00 becomes 1.3 x 10⁷, again, the resulting formula, i.e scientific notation for this number is not describing the actual process to get the number converted, but rather the formula that reverts this process.

Now, I'm trying but still haven't wrapped my head around this: why negative exponents are performing successive divisions?
Why aren't exponentiations always performing multiplications?
I don't know if it's only bad memory but I always remembered exponentiation exclusively as an abbreviated form of writing successive multiplications of a number by itself, never as divisions.