Why is anything to the zero power equal to 1

What happens when we add no numbers at all? Well, we'd expect to get zero, because we're not adding anything at all. But zero is a very special number in addition: it's called the additive identity, because it's the only number which you can add to any other number and leave the other number the same.

So, by this reasoning, it makes sense that if adding no numbers at all gives back the additive identity, multiplying no numbers at all should give the multiplicative identity. Now, what's the multiplicative identity? Well, it's the only number which can be multiplied by any other number without changing that other number.

So, the reason that any number to the zero power is one is because any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1. It's exciting to me that you asked this question. The fact is these rules are presented as somewhat arbitrary, but there is always, always well almost always good reason for them.

Keep it up! If it ever sounds arbitrary then hound your teacher. If your teacher can't give you compelling reasons why something is true, hound us or hound Google. Okay, enough, onto your question:. Mathematics was initially developed to describe relationships between everyday quantities generally whole numbers so the best way to think about powers like a b 'a' raised to the 'b' power is that the answer represents the number of ways you can arrange sets of 'b' numbers from 1 to 'a'.

For example, 2 3 is 8. What would two to the first power be? Well, that would be one, and we would multiply it by one two, one times two, which, of course, is equal to two. Now let's ask ourself an interesting question. Based on this definition of what an exponent is, what would two to the zeroth power be?

I encourage you to just think about that a little bit. If you were the mathematics community, how would you define two to the zero power so it is consistent with everything that we just saw. Well, the way we just talked about it, we just said exponentiation is you start with a one and you multiply it by the base zero times, so we're not gonna multiply it by any two, so we're just gonna be left with a one. So does this make sense that two to the zero power is equal to one?

Well, let's think about it another way, and let's do a different base. That was with two, but let's say we have three and I could say three to the fourth power, that's three times three times three times three which is going to be equal to 81, and let me just write down that this is going to be equal to If I said three to the third power, that's three times three times three which is However, in the case of -1 0 , the negative sign does not signify the number negative one, but instead signifies the opposite number of what follows.

So we first calculate 1 0 , and then take the opposite of that, which would result in Another example: in the expression - -3 2 , the first negative sign means you take the opposite of the rest of the expression.

Why does zero with a zero exponent come up with an error?? Please explain why it doesn't exist. In other words, what is 0 0? Answer: Zero to zeroth power is often said to be "an indeterminate form", because it could have several different values. But we could also think of 0 0 having the value 0, because zero to any power other than the zero power is zero.

So laws of logarithms wouldn't work with it. So because of these problems, zero to zeroth power is usually said to be indeterminate. However, if zero to zeroth power needs to be defined to have some value, 1 is the most logical definition for its value. Google Classroom Facebook Twitter. Video transcript What I want to do in this video is think about exponents in a slightly different way that will be useful for different contexts and also go through a lot more examples.

So in the last video, we saw that taking something to an exponent means multiplying that number that many times. So if I had the number negative 2 and I want to raise it to the third power, this literally means taking three negative 2's, so negative 2, negative 2, and negative 2, and then multiplying them.

So what's this going to be? Well, let's see. Negative 2 times negative 2 is positive 4, and then positive 4 times negative 2 is negative 8. So this would be equal to negative 8. Now, another way of thinking about exponents, instead of saying you're just taking three negative 2's and multiplying them, and this is a completely reasonable way of viewing it, you could also view it as this is a number of times you're going to multiply this number times 1.

So you could completely view this as being equal to-- so you're going to start with a 1, and you're going to multiply 1 times negative 2 three times. So this is times negative 2 times negative 2 times negative 2. So clearly these are the same number. Here we just took this, and we're just multiplying it by 1, so you're still going to get negative 8.

And this might be a slightly more useful idea to get an intuition for exponents, especially when you start taking things to the 1 or 0 power. So let's think about that a little bit. What is positive 2 to the-- based on this definition-- to the 0 power going to be equal to?