What does anything to the zero power equal

Basically, if you have aa this equals 1. You can raise this to any power you want ,. In mathematics, you try to make everything as consistent as possible. If a function works for positive numbers , how might it work for negative numbers? For zero? Let's extend the exponent function from the positives down to 0. It appears that next number should be 1, since that each number is the previous one divided by 2.

If we try this for more numbers, we get the same pattern. All of them result in 1 except for 0, but that is a different story. We use an exponent identity. This one is called the Product Rule.

So we have proved our finding! 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. There are 8 ways to write sets of 3 numbers where each number can be either 1 or 2: 1,1,1 1,1,2 1,2,1 2,1,1 2,1,2 2,2,2 1,2,2 2,2,1. So what does 3 0 represent? It is the number of ways you can arrange the numbers 1,2, and 3 into lists containing none of them! How many ways are there to place a penny, a nickel, and a quarter on the table such that no coins are on the table?

Just one I know this sounds a little fishy since we started with a rule I could have just made up which is why I gave the other reason first , but these formulas are all consistent and there is never any magic step, I promise! One rule for exponents is that exponents add when you have the same base. Now, remember that if you have a negative exponent, it means you have one divided by the number to the exponent:.

If you had trouble understanding it all with variables, let's look at it again,but this time as an example with numbers:. Let's look at what it means to raise a number to a certain power: it means to multiply that number by itself a certain number of times. Let's look at a few examples:. If you look at the pattern, you can see that each time we reduce the power by 1 we divide the value by 3. Using this pattern we can not only find the value of 3 0 , we can find the value of 3 raised to a negative power!

Here are some examples:. No matter what number we use when it is raised to the zero power it will always be 1. Suppose instead of 3 we used some number N, where N could even be a decimal. Heres a quick demonstration of why any number except zero raised to the zero power must equal 1.

As an example we will let that any number be the number 3. 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?

Well, we just said. This says how many times are going to multiply 1 times this number? So this literally says, I'm going to take a 1, and I'm going to multiply by 2 zero times. Well, if I want to multiply it by 2 zero times, that means I'm just left with the 1.

So 2 to the zero power is going to be equal to 1. And, actually, any non-zero number to the 0 power is 1 by that same rationale.

And I'll make another video that will also give a little bit more intuition on there. That might seem very counterintuitive, but it's based on one way of thinking about it is thinking of an exponent as this. And this will also make sense if we start thinking of what 2 to the first power is. So let's go to this definition we just gave of the exponent.

We always start with a 1, and we multiply it by the 2 one time. So 2 is going to be we're only going to multiply it by the 2. I'll use this for multiplication. I'll use the dot. We're only going to multiply it by 2 one time. So 1 times 2, well, that's clearly just going to be equal to 2.