# why is a number to the zero power one Answer 1: This is an excellent question! There are lots
of different ways to think about it, but here's one: let's go back and think about what a power means. When we raise a number to the nth power, that really means that we multiply that number by itself n times, so for example, 2 = 2*2 = 4, 2 = 2*2*2 = 8, 3 = 3*3*3*3 = 81, and so on. So when we raise a number to the zeroth power, that means we multiply the number by itself zero times - but that means we're not multiplying anything at all! What does that mean? Well, let's go even farther back to the simplest case: addition. 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. In short, 0 is the only number such that for any number x, x + 0 = x. 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. In short, the multiplicative identity is the number 1, because for any other number x, 1*x = x. 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. underneath (to create the fraction), I omitted it, because I had the variable expression underneath, and the "times one" doesn't change anything. x / x using only positive exponents. 2 x using only positive exponents. x ", so only the x moves. (3 x using only positive exponents. Unlike the previous exercise, the parentheses meant that the negative power did indeed apply to the three as well as the variable. (-5 x )/( y using only positive powers. ( x / y using only positive exponents. Instead of flipping twice, I noted that all the powers were negative, and moved the outer power onto the inner ones; since "minus times minus is plus", I ended up with all positive powers. Note: While this second solution would be a faster way of getting the exercise done, "faster" doesn't mean "more right". Either way is fine. Since exponents indicate multiplication, and since order doesn't matter in multiplication, there will often be more than one sequence of steps that will lead to a valid simplification of a given exercise of this type. Don't worry if the steps in your homework look quite different from the steps in a classmate's homework. As long as your steps were correct, you should both end up with the same answer in the end. By the way, now that you know about negative exponents, you can understand the logic behind the "anything to the power zero" rule: Why is this so?

There are various explanations. One might be stated as "because that's how the rules work out. " Another would be to trace through a progression like the following: At each stage, with each stage having a power than was one less than what came before, the simplified value was equal to the previous value, divided by. Then logically, since, we must then have:. since anything divided by itself is just " ". There are at least two ways of looking at this quantity: As far as I know, the "math gods" have not yet settled on a firm "definition" of though, to be fair, and just about any programming language will. In calculus, " " will be called an "indeterminate form", meaning that, mathematically, it makes no sense and tells you nothing useful. If this quantity comes up in your class, don't assume: ask your instructor what you should do with it. For loads more worked examples, try. Or continue with this lesson; comes next.

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