how does one "prove" this algebraic rule...?

This is one of those parts of math that falls into the vague territory between a proof and a definition. What you want to show is that the given definition for a^(1/n) "makes sense."

To phrase this more rigorously, we want to show that this definition is consistent with the other rules we have for exponents. Specifically, we have the following rules for integer powers of a:

a^m * a^n = a^(m+n)

(a^m)^n = a^(mn)

a^1 = a

a^0 = 1 (if a≠0)

All of these can be proved for positive integer powers m and n using the definition of a power of a, that is, multiplying a by itself however many times.

Now, we want to figure out what should happen when you take a^(1/n), if n is an integer. Well, our second rule above says that

(a^(1/n))^n = a^(n * 1/n) = a^1 = a

So, in order to make sure that the rules we have still work, a^(1/n) needs to be a number whose nth power is a. Hence, it has to be the nth root of a.

hope that helps!

PS: you can do the same thing with the first rule (rather than the second) to show that

a^-m = 1/(a^m)

when a≠0

For n <> 0 and x > 0, the nth root of x is equal to the positive value rendered by x^(1/n)

See, for example sqrt(4) = 2, but 4^(1/2) can be evaluated as either 2 or -2, so we have to define that we are talking only about the positive result.

So

[x^(1/n)]^n = x^[(1/n) * n] = x^1 = x

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