for anyone curious, here’s a “constructive” explanation of why a⁰ = 1. i’ll also include a “constructive” explanation of why rational exponents are defined the way they are.

anyways, the equality a⁰ = 1 is a consequence of the relation

a^m+1 = a^m • a.

to make things a bit simpler, let’s say a=2. then we want to make sense of the formula

2^m+1 = 2^m • 2

this makes a bit more sense when written out in words: it’s saying that if we multiply 2 by itself m+1 times, that’s the same as first multiplying 2 by itself m times, then multiplying that by 2. for example: 2³ = 2² • 2, since these are just two different ways of writing 2 • 2 • 2.

setting 2⁰ is then what we have to do for the formula to make sense when m = 0. this is because the formula becomes

2⁰⁺¹ = 2⁰ • 2¹.

because 2⁰⁺¹ = 2 and 2¹ = 2, we can divide both sides by 2 and get 1 = 2⁰.

fractional exponents are admittedly more complicated, but here’s a (more handwavey) explanation of them. they’re basically a result of the formula

(a^m)ⁿ = a^m•n

which is true when m and n are whole numbers. it’s a bit more difficult to give a proper explanation as to why the above formula is true, but maybe an example would be more helpful anyways. if m=2 and n=3, it’s basically saying

(a²)³ = (a • a)³ = (a • a) • (a • a) • (a • a) = a^2•3.

it’s worth noting that the general case (when m and n are any whole numbers) can be treated in the same way, it’s just that the notation becomes clunkier and less transparent.

anyways, we want to define fractional exponents so that the formula

(a^r)^s = a^r • a^s

is true when r and s are fractional numbers. we can start out by defining the “simple” fractional exponents of the form a^1/n, where n is a whole number. since n/n = 1, we’re then forced to define a^1/n so that

a = a^1/n•n = (a^1/n)ⁿ.

what does this mean? let’s consider n = 2. then we have to define a^1/2 so that (a^1/2)² = a. this means that a^1/2 is the square root of a. similarly, this means that a^1/n is the n-th root of a.

how do we use this to define arbitrary fractional exponents? we again do it with the formula in mind! we can then just define

a^m/n = (a^1/n)^m.

the expression a^1/n makes sense because we’ve already defined it, and the expression (a^1/n)^m makes sense because we’ve already defined what it means to take exponents by whole numbers. in words, this means that a^m/n is the n-th square root of a, multiplied by itself m times.

i think this kind of explanation can be helpful because they show why exponents are defined in certain ways: we’re really just defining fractional exponents so that they behave the same way as whole number exponents. this makes it easier to remember the definitions, and it also makes it easier to work with them since you can in practice treat them in the “same way” you treat whole number exponents.