I’m too dumb to get it
In: Mathematics
Exponents of integers is just a shorthand notation for a number multiplied by itself a number of times.
So we define the following rules
b^(1) =b
b^(n+1) = b^(n) * b
That mean that bn = b*b*b*b… (you should have n b:s on the right side.)
From that and associativity of multiplication ie that a*(b*c) =(a*b)*c =a*b*c the following line follows if m and n are positive integers.
b^(m+n) =b^(m) * b^(n)
That is the fundamental rule of positive integer exponent. If you look at what happen if the exponent is 0 you can use the face that 4=3+1=4+0 so b to the power of 4 should be identical regardless of how we write it.
b^(4)=b*b*b*b
b^(3+1) = b^(3) * b^(1) =(b*b*b)* b= b*b*b*b
b^(4+0)=b^(4) * b^(0) = (b*b*b*b) * b^(0)
So becaus b^(4) should be the same as b^(4+0) you get the relationship
b^(4) = b^(4+0) => b*b*b*b =(b*b*b*b) * b^(0) => b*b*b*b/(b*b*b*b) = b^(0) =>1=b^(0) So b^(0) has to be 1 for the rules to make sense. That is for any b ≠0 because we divided by b in the equation. what you define 0^(0) as depend on the context.
For the arithmetic operation we have agreed on b^(0) has to be 1 if b≠0 if not the rules do not work. Why for rules that look initially strange in math is generally because that is needed for the rules we have defined to work as smooth and simple as possible. You might say that b^(0) is not allowed but because we can define is as b^(0)=1 and it works fine with other rules we use that definition.
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