Why don't parentheses matter in this case of multipication

Question

Very basic question but can't seem to wrap my head around why this happens.

Normally parentheses indicate that the operation inside must be carried out first.

In this case:

(a * a * a)*(a * a * a * a) = a^7

how does it become a * a * a * a * a * a * a and not (a^3)*(a^4)

I'm looking for an explanation that breaks down the operation into the same way basic exponents and multiplication are explained.

For example 3*3 is explained as 3 cookies being added 3 times. 3*3 = 3+3+3 or 3^2 = 3*3 = 3+3+3

Answer 1

Multiplication is proven to be associative; that means you can calculate any part of a series of it first (= setting and removing parenthesis wherever you want), and the result is always the same.

All the different variants you noted are correct, and they are all identical.

Answer 2

There's an implied exponent of 1 on $a$. That is, $a = a^1$

If you multiply $a * a$, you get $a^1 * a^1 = a^{1+1} = a^2$. When you multiply two powers of the same number, you add their exponents.

So when you multiply $(a*a*a)*(a*a*a*a)$, you're really doing $$(a^1 *a^1 *a^1) * (a^1 * a^1 * a^1 * a^1) = (a^{1+1+1})*(a^{1+1+1+1}) = a^3 *a^4 = a^{3+4} = a^7$$

Answer 3

Another way of thinking about it:

When you have yet to do any multiplying upon what in the future will be a product, you have the proverbial "empty product," which is always $1\quad$. (When you are at that point, your accumulated exponent is $0\quad$.)

When you have yet to do any adding upon what in the future will be a sum, you have the proverbial "empty sum," which is always $0\quad$.

Answer 4

For this special case, it follows from the commutative property of addition, $ a^7 = a^{3 + 4} = a^{4 + 3} $.

In general, multiplication is an associative operation meaning that $ a(bc) = (ab)c$, i.e. you can evaluate in any arbitrary order without affecting the end result.