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LostInParadise's avatar

Is this a good explanation for why a number raised to the zero power equals one?

Asked by LostInParadise (32034points) April 5th, 2022

I think this is something that everyone feels is a bit strange. One approach would be to talk about how exponents are added when you multiply, and that if one number has a zero exponent the answer will have the same exponent as the other number, so multiplying by the zero-exponent number is equivalent to multiplying by 1. That seems overly complicated. What is needed is a more intuitive approach that, at the very least, makes this plausible.

Let’s work with base 2, to make things more concrete. 21 = 2, 22 = 2×2 = 4, 23= 2×2x2=8. Each time the exponent is raised by 1, the previous number is doubled. This is the same as saying that each time the exponent is decreased by 1, the previous value is cut in half. If would then follow that in going from 21 to 2^0, we get 2/2 = 1. The same would hold if we chose any base other than 2.

The argument is not intended to be a knockdown proof, but a suggestion that it is not totally off the wall to conjecture that a number to the zero power is 1.

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9 Answers

JLeslie's avatar

For me, I think math is easier when you just accept the rules. Sometimes it starts to make logical sense later, the more advanced you get. The reason I was good at math K-12 was because I didn’t worry too much about understanding it, I just did what I was told and practiced it. Take order of operations for example, you just need to accept multiplication is before addition, you don’t need to understand why.

The why starts to become more important in more advanced math when trying to figure out how to write (set up) a problem in mathematical terms, but solving once written is more memorization of how to solve.

Your 2/2 explanation makes sense to me, but honestly until now, I had no idea that 2^0 meant to divide the number by itself, or I don’t remember it, but I did know to the power of 0 equals 1.

I’ll send this to LuckyGuy.

zenvelo's avatar

A way to approach it is:

xy /xy = 1
xy /xy = xy-y =x0 =1

LostInParadise's avatar

@JLeslie, I appreciate your answer, but it annoys me that math is taught as a bunch of rules. There is a beauty in math that is not shown.

The order of operation rule was set up as a convenience. There is nothing in mathematics that tells in what order operations should be performed. You could get by just using parentheses. Having a default order of operations allows for fewer parentheses, which makes for easier readability.

@zenvelo , Look at all the steps you had to go through.

zenvelo's avatar

@LostInParadise But my answer in three steps applies to any exponent and any number, with less explanation than your three examples and then wordy explanation for one base number.

RocketGuy's avatar

@zenvelo was thinking of fractions:
24 / 22 = 2(4–2) = 22
22 / 22 = 2(2–2) = 20
22 / 24 = 2(2–4) = 2-2

KRD's avatar

10 to the first power is 10. Anything to the first power is going to be the same number due to that number is the first of the many powers. Anything that is to the zero power is zero because it is like multiplying that number to zero.

zenvelo's avatar

@KRD ”...multiplying that number to zero. Multiplying a number times zero = zero, not one.

zenvelo's avatar

My answer had carets in it to denote exponents that seem to have been stripped when I published.

JLeslie's avatar

@LostInParadise I agree about the beauty, but also keep in mind you have a head for math so I’m guessing the numbers whirl around in your head differently than people who struggle. That’s my experience anyway.

If I say to you to figure 20% tip just take the amount and double it and move the decimal, for instance the bill is $33.00 so $6.60 tip, that’s easy for you, but they get stuck on “double it? But, it’s not double, that would be more.” They try to understand every part in a literal way in my experience. It slows them down.

Take 2+3. You know that is 5 without counting, it just simply is 5 because you know it so well. You have it memorized. I equate that to your number to the zero power. Good to have it explained initially, but eventually it’s just rote.

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