Exponents ≥ 1 are powers; those < 1 are roots. For example:

5^2.3 = (5^2)(5^.3) = (25)[5^(3/10)] = (25)(tenth root of 5^3)

≈ 40.51641

Your question was about X^0, so let’s just look at exponents < 1 in the limit as the exponent goes to 0. For example:

5^.1 = 5^(1/10) = the tenth root of 5, or the number which must be multiplied times itself ten times to equal 5 ≈ 1.17642

5^.01 = 5^(1/100) = the hundredth root of 5 ≈ 1.01622

5^.001 = 5^(1/1000) = the thousandth root of 5 ≈ 1.00161

5^.0001 = the ten-thousandth root of 5 ≈ 1.000161

5^.00001 = the hundred-thousandth root of 5 ≈ 1.0000161

5^.000001 = the millionth root of 5 ≈ 1.00000161

As the exponent gets very small, the order of the root gets very large, and only a number very close to 1 can be multiplied by itself many times to equal a finite number. Slightly larger, and the product goes to infinity; slightly smaller and it goes to 0. Hence, 1 is the infinite-order root of every number. Even for a large number, as the exponent approaches 0, the root quickly approaches 1:

1000000^.00001 ≈ 1.00014

Perhaps one way to visualize X^0 = 1 is to picture each one/single object as the infinite product of all other “similar” objects –

1 grain of sand = 1 grain of sand times 1 grain of sand times 1 grain of sand times …

X^0 = 1 is the answer to the question “What number can be multiplied times itself an infinite number of times and give a finite, nonzero product?”

Physical applications of powers are easier:

4^3 means multiply (4)(4)(4). (4)(4) means, starting with 0, add 4 four times, giving 16. (16)(4) means, starting with 0, add 16 four times (or add 4 sixteen times). The result is 64 (which could be, for example, the volume of a cube with sides of magnitude 4).

Roots (fractional exponents) do not multiply the base number times itself. Rather, they determine another number – the root – which, multiplied times itself gives the base number:

X^(1/n) = the nth root of X

X^(1/n) times X^(1/n) times X^(1/n) …n times = X^1.

If 1/n = 0, then n = ∞ and

X^(1/∞) times X^(1/∞) times X^(1/∞) times …infinite times = 1.