Question
Why does a number raised to the zeroth power equal one?
Answer
Of the mathematical explanations, I like Steve Denton 's best. But none of them is really an "explanation" in the sense that the question is asking for, I think. The question is looking for the "aha!" insight, not the technical reason for correctness. Steve's explanation indirectly shows how the thing SHOULD behave for consistency with the additivity property, but it still doesn't get at why.
There are many things in math (including fairly complicated things, like the Pythagoras theorem) that can be understood through a series of intuitive "aha!" steps that end in one cumulative "aha!"
But there are also many things that seem to elude intuition. Sometimes the reason is obvious (like our fundamental hardware level inability to think in more than 3 dimensions). Other times it is not obvious why something that you can see formally is so difficult to see intuitively. This is one of those things.
Here things are unintuitive because it is not clear what it means to multiply something by itself 0 times. Our basic intuitions about multiplication involve the natural numbers. Even negative exponentiation is actually tricky to grasp, which is why schools teach it at a young age so the kid gets easily distracted and doesn't ask too many questions. So for better intuition, we need an explanation involving only positive exponentiation.
Edwin Chen offers a geometric model, but "zero dimensions" just moves the murkiness from integer arithmetic to geometry.
Matthew Handy is on a more promising track: he is looking at continuity/convergence properties. But he jumps immediately from +1 to -1. That's too coarse.
Let's try to extend this intuition. What about fractional powers? They are roots, right? so [math]x^{\frac{1}{n}}[/math] is the number which, if multiplied by itself n times, gives you x.
So you can think like so:
[math]x^0= \lim_{n\rightarrow \infty} x^{\frac{1}{n}}[/math]
i.e as the limit point of the series:
[math]x^{\frac{1}{1}}, x^{\frac{1}{2}},x^{\frac{1}{3}}, x^{\frac{1}{4}},\ldots[/math]
Two examples: if x=2, you get
2, 1.414, 1.259, 1.189,...
If x=0.5, you get the reciprocals
0.5, 0.707, 0.793, 0.841...
This is still not satisfactory for us intuitive types, but at least you've got a countably infinite series that approaches 1 from both sides of the quantity being raised to a power, and the x=1 case is thankfully intuitive (it's all 1s). So you can get arbitrarily close to [math]x^0[/math] from either side of x, and from the positive direction of the power.
The reason this is not satisfactory is that our intuitions about continuity arguments rely on our geometric sense of the real line. Countably infinite sequences don't cut it (and for good reason: think about [math]\sin{\frac{1}{x}}[/math]).
But thinking in terms of exponentiation of true reals is a) hairy, b) requires more calculus than I care to typeset here and c) will make things more murky, not less.
As you'll find if you dive too deeply into this stuff, the real line is like Alice's Wonderland. If you think the late rabbit that dragged you there is a strange creature, wait till you see the even weirder crap lurking there.
That, then, is the true route to making [math]x^0=1[/math] seem intuitive: juxtapose it against even weirder stuff.
There are many things in math (including fairly complicated things, like the Pythagoras theorem) that can be understood through a series of intuitive "aha!" steps that end in one cumulative "aha!"
But there are also many things that seem to elude intuition. Sometimes the reason is obvious (like our fundamental hardware level inability to think in more than 3 dimensions). Other times it is not obvious why something that you can see formally is so difficult to see intuitively. This is one of those things.
Here things are unintuitive because it is not clear what it means to multiply something by itself 0 times. Our basic intuitions about multiplication involve the natural numbers. Even negative exponentiation is actually tricky to grasp, which is why schools teach it at a young age so the kid gets easily distracted and doesn't ask too many questions. So for better intuition, we need an explanation involving only positive exponentiation.
Edwin Chen offers a geometric model, but "zero dimensions" just moves the murkiness from integer arithmetic to geometry.
Matthew Handy is on a more promising track: he is looking at continuity/convergence properties. But he jumps immediately from +1 to -1. That's too coarse.
Let's try to extend this intuition. What about fractional powers? They are roots, right? so [math]x^{\frac{1}{n}}[/math] is the number which, if multiplied by itself n times, gives you x.
So you can think like so:
[math]x^0= \lim_{n\rightarrow \infty} x^{\frac{1}{n}}[/math]
i.e as the limit point of the series:
[math]x^{\frac{1}{1}}, x^{\frac{1}{2}},x^{\frac{1}{3}}, x^{\frac{1}{4}},\ldots[/math]
Two examples: if x=2, you get
2, 1.414, 1.259, 1.189,...
If x=0.5, you get the reciprocals
0.5, 0.707, 0.793, 0.841...
This is still not satisfactory for us intuitive types, but at least you've got a countably infinite series that approaches 1 from both sides of the quantity being raised to a power, and the x=1 case is thankfully intuitive (it's all 1s). So you can get arbitrarily close to [math]x^0[/math] from either side of x, and from the positive direction of the power.
The reason this is not satisfactory is that our intuitions about continuity arguments rely on our geometric sense of the real line. Countably infinite sequences don't cut it (and for good reason: think about [math]\sin{\frac{1}{x}}[/math]).
But thinking in terms of exponentiation of true reals is a) hairy, b) requires more calculus than I care to typeset here and c) will make things more murky, not less.
As you'll find if you dive too deeply into this stuff, the real line is like Alice's Wonderland. If you think the late rabbit that dragged you there is a strange creature, wait till you see the even weirder crap lurking there.
That, then, is the true route to making [math]x^0=1[/math] seem intuitive: juxtapose it against even weirder stuff.