When using the Product of Powers with equal bases property, why do add the exponents?
Because if you have a number with and exponent(2ex3) and you multiply it by another number with an exponent (2ex5) you are really saying the number x amount of times multiplied by the same number y amoount of times so you have the number x+y amount of times (2ex32ex5=22222222=2ex8)
When using the Quotient of Powers with equal bases Property why do you subtract the exponents?
Because every exponent that you have another one cancels with it. So if you have 2ex4 over 2ex2 you have 2222 over 22 so that means that 2 of the 2 on either side cancel out with each other so you have 2ex2 as the remainder.
When using power of a power property, why do you multiply the exponents?
Because you are essentially doing the number with the exponent a certain amount of times. For example if you have (3ex2)ex3 you really have 3ex23ex23ex2 so 333333 so 3ex6 (3ex2*ex3)
How do negative exponents result and how do we make sense of them
Negative exponents result 1 over the number ex. z. the explanation behind this is when people first had number like 3ex4 over 3ex6 they realized that by canceling 3ex2 really meant 3ex-2 or 1 over 3ex2
How do zero exponents resuly and how do we make sense of them?
When you have a number over a number like 3ex4 over 3ex4 and you use the quotient of properties with equal bases property you get 3ex0 and since any number over itself equals 1 the answer of any number to the 0 power must be one