Where do the rules about how exponentation should be done come from?

Question

Since I first learned the concept of exponents and powers we know that it is "repeated multiplication of number by itself n times" but it breaks apart when I try to rationalize why 0 to power of 0 is = to 1.

I know the answer and many possible explanations tho I do not know the origin of those explanations.

 

In particular this one:

Mathematics was initially developed to describe relationships between everyday quantities (generally whole numbers) so the best way to think about powers like ab ('a' raised to the 'b' power) is that the answer represents the number of ways you can arrange sets of 'b' numbers from 1 to 'a'. For example, 2 to the power of 3 is 8. Why? There are 8 ways to write sets of 3 numbers where each number can be either 1 or 2:(1,1,1) (1,1,2) (1,2,1) (2,1,1) (2,1,2) (2,2,2) (1,2,2) (2,2,1).

So what does 30 represent? It is the number of ways you can arrange the numbers 1,2, and 3 into lists containing none of them! How many ways are there to place a penny, a nickel, and a quarter on the table such that no coins are on the table? Just one... don't put anything on the table and that's your only option. Therefore it's consistent to say 30 = 1.

 

THE QUESTION IS: Why exponentation means the number of times set of numbers can be arranged without repeating?

Answer 1

You are making exponentiation more complicated than it is. Raising a number to a power is just repeated multiplication.

3⁴ means that 3 is to be used 4 times as a factor, and you multiply factors together

3⁴=3*3*3*3=81

5³=5*5*5=125 (5 is used 3 times as a factor)

0⁰ does not exit in mathematics

you can raise a non-zero number to the power of 0 and it is equal to 1

10⁰=1 WHY???

 

10⁴/10³=10^(4-3)=10¹=10

10⁵/10⁵=10^(5-5)=10⁰

but 10⁵/10⁵=100,000/100,000=1

therefore 10⁰=1